Rs Aggarwal 2015 for Class 10 Math Chapter 1

Question 1:

Show that the progression 3, 9, 15, 21,... is an AP. Write down its first term and the common difference.

Answer:

The given progression is 3, 9, 15, 21,...

Clearly, (9 - 3) = (15 - 9) = (21 - 15) = 6 (Constant)
Thus, each term differs from its preceding term by 6.
So, the given progression is an AP.
Its first term is 3 and the common difference is 6.
Thus, a = 3, and d = 6

Question 2:

Show that the progression 16, 11, 6, 1, −4,... is an AP.
Write down its first term and the common difference.

Answer:

The given progression is 16, 11, 6, 1, -4,...
Clearly, (11 - 16) = (6 - 11) = (1 - 6) = (-4 - 1) = -5 (Constant)

Thus, each term differs from its preceding term by -5.
So, the given progression is an AP.
Its first term, a is 16 and common difference, d is -5.

Question 3:

Find:
(i) The 20th term of the AP 1, 5, 9. 13, 17,...
(ii) The 35th term of the AP 6, 9, 12, 15,...
(iii) The nth term of the AP 5, 11, 17, 23,...
(iv) The 11th term of the AP (5a − x), 6a, (7a + x),....

Answer:

(i) The given AP is 1, 5, 9, 13, 17,...
First term, a = 1 and common difference, d = (5 - 1) = 4
T₂₀= a + (20 - 1)d = a + 19d

      = 1 + 19 ⨯ 4 = 77
∴ 20^th term = 77

(ii) T₃₅ = a + (35 - 1)d = a + 34d
= 6 + 34 ⨯ 3 = 108 [∵ a = 6 and d = 3]
∴ 35^th term = 108

(iii) T_n = a + (n - 1)d
= 5 + (n - 1) ⨯ 6
   = 5 + 6n - 6 = 6n - 1 [∵ a = 5 and d = 6]
   ∴ n^th term = 6n - 1

(iv) T₁₁ = a + (11 - 1)d = a + 10d
= 5a - x + 10 ⨯ (a + x) [∵ a = (5a -x) and d = {6a -(5a - x)} = a + x]
= 5a - x + 10a + 10x = 15a + 9x
   ∴ 11^th term = 15a + 9x

Question 4:

Find:
(i) The 10th term of the AP 63, 58, 53, 48,...
(ii) The 14th term of the AP 9, 5, 1, −3,...
(iii) The nth term of the AP 16, 9, 2, −5,...

Answer:

(i) The given AP is 63, 58, 53,48,...
First term, a = 63 and common difference, d = (58 - 63) = -5
T₁₀= a + (10 - 1)d = a + 9d

= 63 + 9 ⨯ (-5) = 63 - 45 = 18
∴ 10^th term = 18

(ii) T₁₄ = a + (14 - 1)d = a + 13d
= 9 + 13 ⨯ (-4) = 9 - 52 = -43 [∵ a = 9 and d = - 4]
∴ 14^th term = - 43

(iii) T_n = a + (n - 1)d
= 16 + (n - 1) ⨯ (-7)          [∵ a = 16 and d = - 7]
           = 16 - 7n + 7 = 23 - 7n
∴ n^th term = 23 -7n

Question 5:

Find the 37th term of the AP $6, 7 \frac{3}{4}, 9 \frac{1}{2}, 11 \frac{1}{4}, . . .$

Answer:

The given AP is $6, 7 \frac{3}{4}, 9 \frac{1}{2}, 11 \frac{1}{4}, . . .$
First term, a = 6 and common difference, d $= 7 \frac{3}{4} - 6 \Rightarrow \frac{31}{4} - 6 \Rightarrow \frac{31 - 24}{4} = \frac{7}{4}$
$Now, T_{37} = a + (37 - 1) d = a + 36 d = 6 + 36 \times \frac{7}{4} = 6 + 63 = 69 ∴ 37^{t h} term = 69$

Question 6:

Find the 25th term of the AP $5, 4 \frac{1}{2}, 4, 3 \frac{1}{2}, 3, . . .$

Answer:

The given AP is $5, 4 \frac{1}{2}, 4, 3 \frac{1}{2}, 3, . . .$
First term = 5
Common difference $= 4 \frac{1}{2} - 5 \Rightarrow \frac{9}{2} - 5 \Rightarrow \frac{9 - 10}{2} = - \frac{1}{2}$
$∴ a = 5 and d = - \frac{1}{2} Now, T_{25} = a + (25 - 1) d = a + 24 d = 5 + 24 \times (- \frac{1}{2}) = 5 - 12 = - 7 ∴ 25^{t h} term = - 7$

Question 7:

How many terms are there in the AP 6, 10, 14, 18,...,174?

Answer:

In the given AP, a = 6 and d = (10 - 6) = 4
Suppose that there are n terms in the given AP.
Then T_n = 174
⇒ a + (n - 1)d = 174
⇒ 6 + (n - 1) ⨯ 4 = 174
⇒ 2 + 4n = 174
⇒ 4n = 172
⇒ n = 43
Hence, there are 43 terms in the given AP.

Question 8:

How many terms are there in the AP 41, 38, 35,...., 8 ?

Answer:

In the given AP, a = 41 and d = (38 - 41) = -3
Suppose that there are n terms in the given AP.
Then T_n = 8
⇒ a + (n - 1) d = 8
⇒ 41 + (n - 1) ⨯ (-3) = 8
⇒ 44 - 3n = 8
⇒ 3n = 36
⇒ n = 12
Hence, there are 12 terms in the given AP.

Question 9:

Which term of the AP 3, 8, 13, 18,... is 88?

Answer:

In the given AP, first term, a = 3 and common difference, d = (8 - 3) = 5.

Let's its n^th term be 88.
Then T_n = 88
⇒ a + (n - 1)d = 88
⇒ 3 + (n - 1) ⨯ 5 = 88
⇒ 5n - 2 = 88
⇒ 5n = 90
⇒ n = 18
Hence, the 18^th term of the given AP is 88.

Question 10:

Which term of the AP 72, 68, 64, 60,... is 0?

Answer:

In the given AP, first term, a = 72 and common difference, d = (68 - 72) = -4.
Let its n^th term be 0.
Then, T_n = 0
⇒ a + (n - 1)d = 0
⇒ 72 + (n - 1) ⨯ (-4) = 0
⇒ 76 - 4n = 0
⇒ 4n = 76
⇒ n = 19
Hence, the 19^th term of the given AP is 0.

Question 11:

Which term of the AP $\frac{5}{6},$ 1, $1 \frac{1}{6}, 1 \frac{1}{3},$ .... is 3?

Answer:

In the given AP, first term = $\frac{5}{6}$ and common difference, d $= (1 - \frac{5}{6} = \frac{1}{6})$ .
Let its n^th term be 3.

$Now, T_{n} = 3 \Rightarrow a + (n - 1) d = 3 \Rightarrow \frac{5}{6} + (n - 1) \times \frac{1}{6} = 3 \Rightarrow \frac{2}{3} + \frac{n}{6} = 3 \Rightarrow \frac{n}{6} = \frac{7}{3} \Rightarrow n = 14$

Hence, the 14^th term of the given AP is 3.

Question 12:

Find the value of x for which (5x + 2), (4x − 1) and (x + 2) are in AP.

Answer:

Let the terms (5x +2), (4x -1) and (x +2) form an AP.

∴ (4x -1) - (5x +2) = (x +2) - (4x -1) [ ∵ Common difference remains constant]
⇒ -x - 3 = -3x + 3
⇒ 2x = 6
⇒ x = 3

Hence, when x = 3, the given terms are in AP.

Question 13:

If the nth term of a progression is (4n − 10), show that it is an AP. Find its
(i) first term, (ii) common difference, and (iii) 16th term.

Answer:

Tn = (4n - 10) [Given]
T₁ = (4 ⨯ 1 - 10) = -6
T₂ = (4 ⨯ 2 - 10) = -2
T₃ = (4 ⨯ 3 - 10) = 2
T₄ = (4 ⨯ 4 - 10) = 6

Clearly, [ -2 - (-6)] = [2 - (-2)] = [6 - 2] = 4 (Constant)

So, the terms -6, -2, 2, 6,... forms an AP.

Thus, we have;
(i) First term = -6

(ii) Common difference = 4
(iii) T₁₆ = a + (n -1)d = a + 15d = - 6 + 15 ⨯ 4 = 54

Question 14:

The 4th and 10th terms of an AP are 13 and 25 respectively. Find the first term and the common difference of the AP. Also, find its 17th term.

Answer:

We have;
T₄ = a + (n - 1)d
⇒ a + 3d = 13 ...(1)

T₁₀ = a + (n - 1)d
⇒ a + 9d = 25 ...(2)
On solving (1) and (2), we get:
a = 7 and d = 2

Thus, we have the following:
(i) First term = 7

(ii) Common difference = 2
(iii) T₁₇ = a + (n - 1)d = a + 16d = 7 + 16 ⨯ 2 = 39

Question 15:

The 8th term of an AP is 37 and its 12th term is 57. Find the AP.

Answer:

We have:
T₈ = a + (n -1)d
⇒ a + 7d = 37 ...(1)

T₁₂ = a + (n -1)d
⇒ a + 11d = 57            ...(2)
On solving (1) and (2), we get:
a = 2 and d = 5

Thus, first term = 2 and common difference = 5

∴ The terms of the AP are 2, 7, 12, 17,...

Question 16:

The 7th term of an AP is −4 and its 13th term is −6. Find the AP.

Answer:

We have:
T₇ = a + (n - 1)d
⇒ a + 6d = -4           ...(1)

T₁₃ = a + (n - 1)d
⇒ a + 12d = -16          ...(2)
On solving (1) and (2), we get:
a = 8 and d = -2

Thus, first term = 8 and common difference = -2

∴ The terms of the AP are 8, 6, 4, 2,...

Question 17:

If the 10th term of an AP is 52 and the 17th term is 20 more than the 13th term, find the AP.

Answer:

In the given AP, let the first term be a and the common difference be d.
Then, T_n = a + (n - 1)d
Now, we have:
T₁₀ = a + (10 - 1)d
⇒ a + 9d = 52       ...(1)
T₁₃ = a + (13 - 1)d = a + 12d               ...(2)
T₁₇ = a + (17 - 1)d = a + 16d                ...(3)

But, it is given that T₁₇ = 20 + T₁₃
i.e., a + 16d = 20 + a + 12d
⇒ 4d = 20
⇒ d = 5
On substituting d = 5 in (1), we get:
a + 9 ⨯ 5 = 52
⇒ a = 7

Thus, a = 7 and d = 5

∴ The terms of the AP are 7, 12, 17, 22,...

Question 18:

The 4th term of an AP is zero. Prove that its 25 term is triple is 11th term.

Answer:

In the given AP, let the first term be a and the common difference be d.
Then, T_n = a + (n - 1)d
Now, T₄ = a + (4 - 1)d
⇒ a + 3d = 0 ...(1)

⇒ a = -3d

Again, T₁₁ = a + (11 - 1)d = a + 10d
= -3d + 10 d = 7d [ Using (1)]

Also, T₂₅ = a + (25 - 1)d = a + 24d = -3d + 24d = 21d [ Using (1)]
i.e., T₂₅ = 3 ⨯ 7d = (3 ⨯ T₁₁)
Hence, 25^th term is triple its 11^th term.

Question 19:

Which term of the AP 3, 8, 13, 18, ... will be 55 more than its 20th term?

Answer:

Here, a = 3 and d = (8 - 3) = 5
The 20^th term is given by
T₂₀ = a + (20 - 1)d = a + 19d = 3 + 19 ⨯ 5 = 98
∴ Required term = (98 + 55) = 153
Let this be the n^th term.
Then T_n = 153
⇒ 3 + (n - 1) ⨯ 5 = 153
⇒ 5n = 155
⇒ n = 31
Hence, the 31^st term will be 55 more than its 20^th term.

Question 20:

Which term of the AP 5, 15, 25, ...will be 130 more than its 31st term?

Answer:

Here, a = 5 and d = (15 - 5) = 10
The 31^st term is given by
T₃₁ = a + (31 - 1)d = a + 30d = 5 + 30 ⨯ 10 = 305
∴ Required term = (305 + 130) = 435
Let this be be the n^th term.
Then T_n = 435
⇒ 5 + (n - 1) ⨯ 10 = 435
⇒ 10n = 440
⇒ n = 44
Hence, the 44^th term will be 130 more than its 31^st term.

Question 21:

For what value of n, the nth terms of the arithmetic progressions 63, 65, 67,... and 3, 10, 17,... are equal?

Answer:

Let the n^th term of the given progressions be t_n and T_n, respectively.
The first AP is 63, 65, 67,...
Let its first term be a and common difference be d.
Then a = 63 and d = (65 - 63) = 2
So, its n^_th term is given by
t_n = a + (n - 1)d
⇒ 63 + (n - 1) ⨯ 2
⇒ 61 + 2n

The second AP is 3, 10, 17,...
Let its first term be A and common difference be D.
Then A = 3 and D = (10 - 3) = 7
So, its n^_th term is given by
T_n = A + (n - 1)D
⇒ 3 + (n - 1) ⨯ 7
⇒ 7n - 4
Now, t_n = T_n⇒ 61 + 2n = 7n - 4
⇒ 65 = 5n
⇒ n = 13
Hence, the 13^th terms of the AP's are the same.

Question 22:

How many 3-digit numbers are divisible by 7?

Answer:

The smallest three-digit number which is divisible by 7 is 105 and the largest three-digit number divisible by 7 is 994.
Hence, the progression is 105, 112, 119,..., 994
Here, a = 105, d = 7 and last term = 994
Let n be the number of terms of the sequence.
Now, we have:
T_n = 994 ⇒ a + (n - 1)d = 994
⇒ 105 + (n - 1) ⨯ 7 = 994
⇒ 7n-7 = 889
⇒ 7n = 896
⇒ n = 128

Hence, there are 128 three-digit numbers which are divisible by 7.

Question 23:

Find the 8th term from the end of the AP 7, 10, 13, ..., 184.

Answer:

Here, a = 7 and d = (10 - 7) = 3, l = 184 and n = 8^th from the end.
Now, n^th term from the end = [l - (n -1)d]
8^th term from the end = [184 - (8 - 1) ⨯ 3]
= [184 - (7 ⨯ 3)] = (184 - 21) = 163
Hence, the 8^th term from the end is 163.

Question 24:

Find the 6th term from the end of the AP 17, 14, 11, ..., (−40).

Answer:

Here, a = 17 and d = (14 - 17) = -3, l = (-40) and n = 6

Now, n^th term from the end = [l - (n - 1)d]
6^th term from the end = [(-40) - (6 - 1) ⨯ (-3)]
= [-40 + (5 ⨯ 3)] = (-40 + 15) = -25
Hence, the 6^th term from the end is -25.

Question 25:

Find the 11th term from the last term towards the first term of the AP 10, 7, 4,.... (−62).

Answer:

Here, a = 10 and d = (7 - 10) = -3, l = (-62) and n = 11

Now, n^th term from the end = [l - (n - 1)d]
11^th term from the end = [(-62) - (11 - 1) ⨯ (-3)]
= [-62 + (10 ⨯ 3)] = (-62 + 30) = -32

Hence, the 11^th term from the last towards the first term of the AP is -32.

Question 26:

If the pth term of an AP is q and its qth term is p then show that its (p + q)th term is zero.

Answer:

In the given AP, let the first term be a and the common difference be d.
Then T_n = a + (n - 1)d
⇒ Tp = a + (p - 1)d = q ...(i)

⇒ T_q = a + (q - 1)d = p ...(ii)

On subtracting (i) from (ii), we get:
(q - p)d = (p - q)
⇒ d = -1
Putting d = -1 in (i), we get:
a = (p + q - 1)

Thus, a = (p + q - 1) and d = -1
Now, T_p₊_q = a + (p + q - 1)d
= (p + q - 1) + (p + q - 1)(-1)

= (p + q - 1) - (p + q - 1) = 0

Hence, the (p+q)^th term is 0 (zero).

Question 27:

If 10 times the 10th term of an AP is equal to 15 times the 15th term, show that its 25th term is zero.

Answer:

In the given AP, let the first term be a and the common difference be d.
Now, T_n = a + (n - 1)d
Then we have:
T₁₀ = a + (10 - 1)d = a + 9d

T₁₅ = a + (15 - 1)d = a + 14d

Now, (10 ⨯ T₁₀) = (15 ⨯ T₁₅)
⇒10[ a + 9d] = 15 [a + 14d ]
⇒ 10a + 90d = 15a + 210d

⇒ -5a = 120d
⇒ a = -24d

∴ T₂₅ = a + (25 - 1)d = a + 24d = -24d + 24d = 0 [ ∵ a = -24d]
Hence, the 25^th term is 0 (zero).

Question 28:

The first and last terms of an AP are a and l, respectively. Show that the sum of the nth term from the beginning and the nth term from the end is (a + 1).

Answer:

In the given AP, first term = a and last term = l.
Let the common difference be d.

Then, n^th term from the beginning is given by
T_n = a + (n - 1)d ...(1)

Similarly, n^th term from the end is given by
T_n = {l - (n - 1)d} ...(2)
Adding (1) and (2), we get:
a + (n - 1)d + {l - (n - 1)d}
= a + (n - 1)d + l - (n - 1)d
= a + l

Hence, the sum of the n^th term from the beginning and the n^th term from the end is (a + l) .

Question 29:

In a flower bed, there are 43 rose plants in the first row, 41 in the second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?

Answer:

The numbers of rose plants in consecutive rows are 43, 41, 39,..., 11.

Difference of rose plants between two consecutive rows = (41 - 43) = (39 - 41) = -2 [Constant]
So, the given progression is an AP.
Here, first term = 43
Common difference = -2
Last term = 11
Let n be the last term, then we have:
T_n = a + (n - 1)d
⇒ 11 = 43 + (n - 1)(-2)
⇒ 11 = 45 - 2n
⇒ 34 = 2n
⇒ n = 17
Hence, the 17^th term is 11 or there are 17 rows in the flower bed.

Question 30:

A sum of Rs 1000 is invested at 8% per annum simple interest. Calculate the interest at the end of each year. Show that these interest form an AP. Using this fact, find the interest at the end of 30 years.

Answer:

Simple interest = \frac{(P \times R \times T)}{100}, where P = principal, R = rate of interest and T = time in years . = \frac{(1000 \times 8 \times T)}{100} = 80 \times T

On putting T = 1, 2, 3, 4, we get:
Rs 80, Rs 160, Rs 240, Rs 320 and so on.

The difference between the interests of two consecutive years = (160 - 80) = (240 - 160) = Rs 80 [Constant]
So, the progression of simple interests forms an AP.
Here, first term = Rs 80
Common difference = Rs 80

Now, interest at the end of 30 years, T₃₀= a + (n - 1)d
= 80 + (30 - 1)(80) = 80 + 29(80) = Rs 2400
Hence, the interest at the end of 30 years is Rs 2400.

Question 1:

Write the value of x for which (x + 2), 2x, (2x + 3) are three consecutive terms of an AP?

Answer:

Since (x + 2), 2x and (2x + 3) are in AP, we have:
2x - (x+2) = (2x+3) -2x
⇒ x - 2 = 3
⇒ x = 5
∴ x = 5

Question 2:

Find three numbers in AP whose sum is 15 and product is 80.

Answer:

Let the required numbers be (a - d), a and (a + d).
Then (a - d) + a + (a + d) = 15
⇒ 3a = 15
⇒ a = 5
Also, (a - d).a.(a + d) = 80
⇒ a(a² - d²) = 80
⇒ 5 (25 - d²) = 80
⇒ d² = 25 - 16 = 9
⇒ d = $\pm$ 3
Thus, a = 5 and d = $\pm$ 3
Hence, the required numbers are (2, 5 and 8) or ( 8, 5 and 2).

Question 3:

The sum of three numbers in AP is 27 and their product is 405. Find the numbers.

Answer:

Let the required numbers be (a - d), a and (a + d).
Then (a - d) + a + (a + d) = 27
⇒ 3a = 27
⇒ a = 9
Also, (a - d).a.(a + d) = 405
⇒ a(a² - d²) = 405
⇒ 9(81 - d²) = 405
⇒ d² = 81 - 45 = 36
⇒ d = $\pm$ 6
Thus, a = 9 and d = $\pm$ 6
Hence, the required numbers are (3, 9 and 15) or (15, 9 and 3).

Question 4:

The sum of three numbers in AP is 3 their product is −35. Find the numbers.

Answer:

Let the required numbers be (a - d), a and (a + d).
Then (a - d) + a + (a + d) = 3
⇒ 3a = 3
⇒ a = 1
Also, (a - d).a.(a + d) = -35
⇒ a(a² - d²) = -35
⇒ 1.(1 - d²) = -35
⇒ d² = 36
⇒ d = $\pm$ 6

Thus, a = 1 and d = $\pm$ 6
Hence, the required numbers are ( -5, 1 and 7) or ( 7, 1 and -5).

Question 5:

Divide 24 in three parts such that they are in AP and their product is 440.

Answer:

Let the required parts of 24 be (a - d), a and (a + d) such that they are in AP.
Then (a - d) + a + (a + d) = 24
⇒ 3a = 24
⇒ a = 8
Also, (a - d).a.(a + d) = 440
⇒ a(a² - d²) = 440
⇒ 8(64 -d²) = 440
⇒ d² = 64 - 55 = 9
⇒ d = $\pm$ 3
Thus, a = 8 and d = $\pm$ 3
Hence, the required parts of 24 are (5, 8,11) or (11, 8, 5).

Question 6:

The sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms.

Answer:

Let the required terms be (a - d), a and (a + d).
Then (a - d) + a + (a + d) = 21
⇒ 3a = 21
⇒ a = 7
Also, (a - d)²+a² + (a + d)² = 165
⇒ 3a² + 2d² = 165
⇒ (3 ⨯ 49 + 2 d²) = 165
⇒ 2d² = 165 - 147 = 18
⇒ d² = 9
⇒ d = $\pm$ 3
Thus, a = 7 and d = $\pm$ 3
Hence, the required terms are ( 4, 7,10) or ( 10, 7, 4).

Question 7:

The angle of a quadrilateral are in AP whose common difference is 10°. Find the angles.

Answer:

Let the required angles be (a - 15)^o, (a - 5)^o, (a + 5)^o and (a + 15)^o, as the common difference is 10 (given).
Then (a - 15)^o + (a - 5)^o + (a + 5)^o + (a + 15)^o = 360^o
⇒ 4a = 360
⇒ a = 90

Hence, the required angles of a quadrilateral are $(90 - 15) °, (90 - 5) °, (90 + 5) ° and (90 + 15) °; or 75 °, 85 °, 95 ° and 105 ° .$

Question 8:

Find your numbers in AP whose sum is 28 and the sum of whose squares is 216.

Answer:

Let the required numbers be (a - 3d)(a - d), (a + d) and (a + 3d).
Then (a - 3d) + (a - d) + (a + d) + (a + 3d) = 28
⇒ 4a = 28
⇒ a = 7

Also, (a - 3d)²+(a - d)²+(a + d)² + (a + 3d)² = 216
⇒ 4 (a² + 5d²)= 216
⇒ 4⨯( 49 + 5d²) = 216
⇒ 5d² = 54 - 49 = 5
⇒ d² = 1
⇒ d = $\pm$ 1
Thus, a = 7 and d = $\pm$ 1
Hence, the required numbers are (4, 6, 8, 10) or (10, 8, 6, 4).

Question 1:

Find the sum of first 19 terms of the AP 2, 7, 12, 17, ..... .

Answer:

Here, a = 2, d = (7 - 2) = 5 and n = 19
Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{19} = \frac{19}{2} [2 \times 2 + (19 - 1) 5] [∵ a = 2, d = 5 and n = 19] = \frac{19}{2} [4 + 90] = 19 \times 47 = 893$
Hence, the sum of the first 19 terms of the given AP is 893.

Question 2:

Find the sum of first 26 terms of the AP 1, 3, 5, 7, ..... .

Answer:

Here, a = 1, d = (3-1) = 2 and n = 26
Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{26} = \frac{26}{2} [1 \times 2 + (26 - 1) 2] [∵ a = 1, d = 2 and n = 26] = 13 \times [2 + 50] = 13 \times 52 = 676$

Hence, the sum of the first 26 terms of the given AP is 676.

Question 3:

Find the sum of first 18 terms of the AP 9, 7, 5, 3, ..... .

Answer:

Here, a = 9, d = (7 - 9) = -2 and n = 18
Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{18} = \frac{18}{2} [2 \times 9 + (18 - 1) \times (- 2)] [∵ a = 9, d = - 2 and n = 18] = 9 \times [18 - 34] = 9 \times (- 16) = - 144$

Hence, the sum of the first 18 terms of the given AP is -144.

Question 4:

Find the sum (5 + 13 + 21 + ... + 181).

Answer:

Here, a = 5, d = (13 - 5) = 8 and l = 181
Let the total number of terms be n.
Then T_n = 181
⇒ a + (n - 1)d = 181
⇒ 5 + (n - 1) ⨯ 8 = 181
⇒ 8n = 184
⇒ n = 23
Hence, there are 23 terms in the AP.
∴ Required sum = $\frac{n}{2} (a + l)$

= $\frac{23}{2} (5 + 181) = 23 \times 93 = 2139$
Hence, the required sum is 2139.

Question 5:

Which term of the AP 5, 9, 13, 17, ... is 81? Also find the sum (5 + 9 + 13 + 17 + ... + 81).

Answer:

In the given AP, first term, a = 5 and common difference, d = (9 - 5) = 4.
Let its n^thterm be 81.
Then T_n = 81
⇒ a + (n - 1)d = 81
⇒ 5 + (n - 1) ⨯ 4 = 81
⇒ 4n = 80
⇒ n = 20
Hence, the 20^th term of the given AP is 81.

∴ Required sum = $\frac{n}{2} (a + l)$
= $\frac{20}{2} (5 + 81) = 10 \times 86 = 860$
Hence, the required sum is 860.

Question 6:

Which term of the AP 15, 11, 7, ... is (−13)? Also find the sum (15 + 11 + 7 + ... + (−13).

Answer:

In the given AP, first term, a = 15 and common difference, d = (11 - 15) = -4.
Let its n^thterm be -13.
Then T_n = -13
⇒ a + (n - 1)d = -13
⇒ 15 + ( n - 1) ⨯ (-4) = -13
⇒ 4n = 32
⇒ n = 8
Hence, the 8^th term of the given AP is -13.
∴ Required sum = $\frac{n}{2} (a + l)$
= $\frac{8}{2} [15 + (- 13)] = 8$
Hence, the required sum is 8.

Question 7:

Find the sum of al two-digit whole numbers divisible by 3.

Answer:

All the two-digit whole numbers divisible by 3 are 12, 15, 18, 21,..., 99.
This is an AP in which a = 12, d = (15 - 12) = 3 and l = 99.
Let the number of terms be n.
Then first term (a) = 12 and common difference (d) = (15 - 12) = 3
Now, T_n = 99
⇒ a + (n - 1)d = 99
⇒ 12 + ( n - 1) ⨯ 3 = 99
⇒ 3n = 90
⇒ n = 30

∴ Required sum = $\frac{n}{2} (a + l)$
= $\frac{30}{2} [12 + 99] = 15 \times 111 = 1665$
Hence, the required sum is 1665.

Question 8:

Find the sum of all even numbers between 5 and 100.

Answer:

All the even numbers between 5 and 100 are 6, 8, 10, 12,..., 98.

This is an AP in which a = 6, d = (8 - 6) = 2 and l = 98.

Let the number of terms be n.
Then T_n = 98
⇒ a + (n - 1)d = 98
⇒ 6 + ( n - 1) ⨯ 2 = 98
⇒ 2n = 94
⇒ n = 47

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{47}{2} [6 + 98] = 47 \times 52 = 2444

Hence, the required sum is 2444.

Question 9:

Find the sum of all natural numbers between 100 and 500 which are divisible by 7.

Answer:

All the natural numbers less than 100 and divisible by 6 are 6, 12, 18, 24,...,96.

This is an AP in which a = 6, d = (12 - 6) = 6 and l = 96.

Let the number of terms be n.
Then T_n = 96
⇒ a + (n - 1)d = 96
⇒ 6 + (n - 1) ⨯ 6 = 96
⇒ 6n = 96
⇒ n = 16

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{16}{2} [6 + 96] = 8 \times 102 = 816

Hence, the required sum is 816.

Question 10:

Find the sum of all natural numbers between 100 and 500 which are divisible by 7.

Answer:

All the natural numbers between 100 and 500 which are divisible by 7 are 105, 112, 117, 126,..., 497.

This is an AP in which a = 105, d = (112 - 105) = 7 and l = 497.

Let the number of terms be n.
Then T_n = 497
⇒ a + (n - 1)d = 497
⇒ 105 + (n - 1) ⨯ 7 = 497
⇒ 7n = 399
⇒ n = 57

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{57}{2} [105 + 497] = 57 \times 301 = 17157

Hence, the required sum is 17157

Question 11:

Find the sum of all multiples of 9 lying between 300 and 700.

Answer:

All the multiples of 9 lying between 300 and 700 are 306, 315, 324, 333, ..., 693.

This is an AP in which a = 306, d = (315 - 306) = 9 and l = 693.

Let the number of terms be n.
Then T_n = 693
⇒ a + (n - 1)d = 693
⇒ 306 + (n - 1) ⨯ 9 = 693
⇒ 9n = 396
⇒ n = 44

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{44}{2} [306 + 693] = 22 \times 999 = 21978

Hence, the required sum is 21978.

Question 12:

Find the sum of all three-digit natural numbers which are divisible by 13.

Answer:

All three-digit numbers which are divisible by 13 are 104, 117, 130, 143,..., 988.

This is an AP in which a = 104, d = (117 - 104) = 13 and l = 988

Let the number of terms be n.
Then T_n = 988
⇒ a + (n - 1)d = 988
⇒ 104 + (n -1) ⨯ 13 = 988
⇒ 13n = 897
⇒ n = 69

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{69}{2} [104 + 988] = 69 \times 546 = 37674

Hence, the required sum is 37674.

Question 13:

Find the sum of first 15 multiples of 8.

Answer:

The first 15 multiples of 8 are 8, 16, 24, 32,...

This is an AP in which a = 8, d = (16 - 8) = 8 and n = 15.
Thus, we have:

l = a + (n - 1) d = 8 + (15 - 1) 8 = 120

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{15}{2} [8 + 120] = 15 \times 64 = 960

Hence, the required sum is 960.

Question 14:

Find the sum of all odd numbers between 0 and 50.

Answer:

All odd numbers between 0 and 50 are 1, 3, 5, 7, ..., 49.

This is an AP in which a = 1, d = (3 - 1) = 2 and l = 49.

Let the number of terms be n.
Then, T_n = 49
⇒a + (n - 1)d = 49
⇒ 1 + (n - 1) ⨯ 2 = 49
⇒ 2n = 50
⇒ n = 25

∴ Required sum =

\frac{n}{2} (a + l)

=

\frac{25}{2} [1 + 49] = 25 \times 25 = 625

Hence, the required sum is 625.

Question 15:

Find the sum of first hundred even natural numbers which are divisible by 5.

Answer:

The first few even natural numbers which are divisible by 5 are 10, 20, 30, 40, ...

This is an AP in which a = 10, d = (20 - 10) = 10 and n = 100

The sum of n terms of an AP is given by

S_{n} = \frac{n}{2} [2 a + (n - 1) d] = (\frac{100}{2}) \times [2 \times 10 + (100 - 1) \times 10] [∵ a = 10, d = 10 and n = 100] = 50 \times [20 + 990] = 50 \times 1010 = 50500

Hence, the sum of the first hundred even natural numbers which are divisible by 5 is 50500.

Question 16:

Find the sum of all 3-digit natural numbers which are divisible by 13.

Answer:

All three-digit numbers which are divisible by 13 are 104, 117, 130, 143, ..., 988.

This is an AP in which a = 104, d = (117 - 104) = 13 and l = 988.

Let the number of terms be n.
Then T_n = 988
⇒ a + (n - 1)d = 988
⇒ 104 + (n - 1) ⨯13 = 988
⇒ 13n = 897
⇒ n = 69

∴ Required sum =

\frac{n}{2} (a + l)

= \frac{69}{2} (104 + 988) = 69 \times 546 = 37674

Hence, the required sum is 37674.

Question 17:

Find the sum of 51 terms of the AP whose second term is 2 and the 4th term is 8.

Answer:

Let a be the first term and d be the common difference of the given AP.
Then T₂ = 2 and T₄ = 8

⇒ a + (2 - 1)d = 2 and a + (4 - 1)d = 8
i.e., a + d = 2    ...(i)
also, a + 3d = 8    ...(ii)
On subtracting (i) from (ii), we get:
2d = 6
⇒ d = 3
Putting d = 3 in (i), we get:
a = -1
∴ a = -1, d = 3 and n = 51

Now, sum of the first 51 terms =

\frac{n}{2} [2 a + (n - 1) d]

= (\frac{51}{2}) \times [2 \times (- 1) + (51 - 1) \times 3] = (\frac{51}{2}) \times [- 2 + 150] = (\frac{51}{2}) \times 148 = 51 \times 74 = 3774

Hence, the required sum is 3774.

Question 18:

If the 5th and 12th terms of an AP −4 and −18 respectively, find the sum of first 20 terms of the AP.

Answer:

Let a be the first term and d be the common difference of the given AP.
Then T₅ = -4 and T₁₂ = -18

⇒a + (5 - 1) d = -4 and a + (12 - 1) d = -18
i.e., a + 4d = - 4 ...(i)
also, a +11d = -18 ...(ii)
On subtracting (i) from (ii), we get:
7d = -14
⇒ d = -2
Putting d = -2 in (i), we get:
a = 4
∴ a = 4, d = -2 and n = 20
Sum of the first 20 terms =

\frac{n}{2} [2 a + (n - 1) d]

= (\frac{20}{2}) \times [2 \times 4 + (20 - 1) \times (- 2)] = 10 \times [8 - 38] = 10 \times (- 30) = - 300

Hence, the required sum is -300.

Question 19:

How many terms of the AP 21, 18, 15, ... must be added to get the sum 0?

Answer:

Here, a = 21 and d = (18 - 21) = -3

Let the required number of terms be n.
Then, S_n = 0
⇒

\frac{n}{2} [2 a + (n - 1) d]

= 0

\Rightarrow (\frac{n}{2}) \times [2 \times 21 + (n - 1) \times (- 3)] = 0 \Rightarrow (\frac{n}{2}) \times [45 - 3 n] = 0 \Rightarrow 45 n - 3 n^{2} = 0 \Rightarrow 3 n = 45 \Rightarrow n = 15

Hence, the required number of terms is 15.

Question 20:

Find the number of terms of the AP 63, 60, 57, ... so that their sum is 693. Explain the double answer.

Answer:

Here, a = 63 and d = (60 - 63) = -3

Let the required number of terms be n.
Then, S_n = 693
⇒

\frac{n}{2} [2 a + (n - 1) d]

= 693

\Rightarrow (\frac{n}{2}) \times [2 \times 63 + (n - 1) \times (- 3)] = 693 \Rightarrow (\frac{n}{2}) \times [129 - 3 n] = 693 \Rightarrow 3 n^{2} - 129 n + 1386 = 0 \Rightarrow n^{2} - 43 n + 462 = 0 \Rightarrow n^{2} - 21 n - 22 n + 462 = 0 \Rightarrow n (n - 21) - 22 (n - 21) = 0 \Rightarrow (n - 21) (n - 22) = 0 \Rightarrow n = 21 o r n = 22

∴ Sum of the first 21 terms = sum of the first 22 terms = 693
This means that the term T₂₂ is 0.

Question 21:

Find the number of terms of the AP 64, 60, 56, ... so that their sum is 544. Explain the double answer.

Answer:

Here, a = 64 and d = (60 - 64) = -4

Let the required number of terms be n.
Then, S_n = 544
⇒

\frac{n}{2} [2 a + (n - 1) d]

= 544

\Rightarrow (\frac{n}{2}) \times [2 \times 64 + (n - 1) \times (- 4)] = 544 \Rightarrow (\frac{n}{2}) \times [132 - 4 n] = 544 \Rightarrow 2 n^{2} - 66 n + 544 = 0 \Rightarrow n^{2} - 33 n + 272 = 0 \Rightarrow n^{2} - 17 n - 16 n + 272 = 0 \Rightarrow n (n - 17) - 16 (n - 17) = 0 \Rightarrow (n - 17) (n - 16) = 0 \Rightarrow n = 17 o r n = 16

∴ Sum of the first 16 terms = sum of the first 17 terms = 544
This means that the term T₁₇ is 0.

Question 22:

Find the number of terms of the AP 18, 15, 12, ... so that their sum is 45. Explain the double answer.

Answer:

Here, a = 18 and d = (15- 18) = -3

Let the required number of terms be n.
Then, S_n = 45
⇒

\frac{n}{2} [2 a + (n - 1) d]

= 45

\Rightarrow (\frac{n}{2}) \times [2 \times 18 + (n - 1) \times (- 3)] = 45 \Rightarrow (\frac{n}{2}) \times [39 - 3 n] = 45 \Rightarrow 3 n^{2} - 39 n + 90 = 0 \Rightarrow n^{2} - 13 n + 30 = 0 \Rightarrow n^{2} - 10 n - 3 n + 30 = 0 \Rightarrow n (n - 10) - 3 (n - 10) = 0 \Rightarrow (n - 10) (n - 3) = 0 \Rightarrow n = 3 o r n = 10

∴ Sum of the first 3 terms = sum of the first 10 terms = 45
This means that the sum of all terms from 4^th to 10^th is zero.

Question 23:

If the nth term of an AP is (4n + 1), find the sum or the first 15 terms of this AP. Also find the sum of its n terms.

Answer:

Given: T_n = (4n + 1).
Now, T₁ = (4 ⨯ 1 + 1) = 5
T₂ = (4 ⨯ 2 + 1) = 9
T₁₅ = (4 ⨯ 15 + 1) = 61
Hence, a = 5, d = (9 - 5) = 4 and l = 61

∴ S₁₅ =

\frac{n}{2} (a + l)

= (\frac{15}{2}) \times (5 + 61) = 15 \times 33 = 495

Hence, the sum of the first 15 terms is 495.
The sum of the first n terms is given by

S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [2 \times 5 + (n - 1) \times 4] = \frac{n}{2} [6 + 4 n] = (3 n + 2 n^{2})

Question 24:

The sum of the first n terms of an AP is given by S_n = (2n² + 5n). Find the nth term of the AP.

Answer:

Given: S_n = (2n² + 5n) ...(i)
Replacing n by (n - 1) in (i), we get:
S_n-1 = 2(n - 1)² + 5(n - 1)
= 2(n² - 2n + 1) + 5n - 5
= 2n² + n - 3
∴ T_n = (S_n - S_n-1)
= (2n² + 5n) - (2n² + n - 3 ) = 4n +3
∴ n^th term, T_n = (4n +3)

Question 25:

If the sum of the first n terms of an AP is given by Sn = (3n² − n), find its (i) nth term, (ii) first term, and (iii) common difference.

Answer:

Given: S_n = (3n² - n) ...(i)
Replacing n by (n - 1) in (i), we get:
S_n-1 = 3(n - 1)² - (n - 1)
= 3(n² - 2n + 1) - n + 1
= 3n² - 7n + 4
(i) Now, T_n = ( S_n - S_n-1)
= (3n² - n) - (3n² - 7n + 4 ) = 6n - 4
∴ n^th term, T_n = (6n - 4) ...(ii)

(ii) Putting n = 1 in (ii), we get:
T₁= (6 ⨯ 1) - 4 = 2

(iii) Putting n = 2 in (ii), we get:
T₂= (6 ⨯ 2) - 4 = 8
∴ Common difference, d = T₂ - T₁ = 8 - 2 = 6

Question 26:

The sum of n terms of an AP is 49 and that of first 17 terms is 289, find the sum of first n terms.

Answer:

$S_{n} = (\frac{5 n^{2}}{2} + \frac{3 n}{2}) = \frac{1}{2} (5 n^{2} + 3 n) . . . (i) Replacing n by (n - 1) in (i), we get : S_{n - 1} = \frac{1}{2} \times [5 {(n - 1)}^{2} + 3 (n - 1)] = \frac{1}{2} \times [5 n^{2} - 10 n + 5 + 3 n - 3]] = \frac{1}{2} \times [5 n^{2} - 7 n + 2] ∴ T_{n} = S_{n} - S_{n - 1} = \frac{1}{2} (5 n^{2} + 3 n) - \frac{1}{2} \times [5 n^{2} - 7 n + 2] = \frac{1}{2} (10 n - 2) = 5 n - 1 . . . (i i)$

Putting n = 20 in (ii), we get:
T₂₀= (5 ⨯ 20) - 1 = 99
Hence, the 20^th term is 99.

Question 27:

If the sum of first 7 terms of an AP is 49 and that of first 17 terms is 289, find the sum of first n terms.

Answer:

Let a be the first term and d be the common difference of the given AP.
Then we have:
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] S_{7} = \frac{7}{2} [2 a + 6 d] = 7 [a + 3 d] S_{17} = \frac{17}{2} [2 a + 16 d] = 17 [a + 8 d]$

However, S₇ = 49 and S₁₇ = 289
Now, 7[a + 3d] = 49
⇒ a + 3d = 7 ...(i)
Also, 17[a + 8d] = 289
⇒ a + 8d = 17 ...(ii)

Subtracting (i) from (ii), we get:

5d = 10
⇒ d = 2

Putting d = 2 in (i), we get:
a + 6 = 7
⇒ a = 1
Thus, a = 1 and d = 2

∴ Sum of n terms of AP =

\frac{n}{2} [2 \times 1 + (n - 1) \times 2] = n [1 + (n - 1)] = n^{2}

Question 28:

The sum of the first 9 terms of an AP is 81 and the sum of its first 20 terms is 400. Find the first term and the common difference of the AP.

Answer:

Let a be the first term and d be the common difference of the given AP.
Then we have:
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] S_{9} = \frac{9}{2} [2 a + 8 d] = 9 [a + 4 d] S_{20} = \frac{20}{2} [2 a + 19 d] = 10 [2 a + 19 d]$

However, S₉ = 81 and S₂₀ = 400

Now, 9[a + 4d] = 81
⇒ a + 4d = 9 ...(i)
Also, 10[ 2a +19d ] = 400
⇒2a + 19d = 40 ...(ii)

Multiplying (i) by 2 and subtracting the result from (ii), we get:

11d = 22
⇒ d = 2

Putting d = 2 in (i), we get:
a + 8 = 9 ⇒a = 1
Thus, a = 1 and d = 2

Question 29:

The first and last terms of an AP are 4 and 81 respectively. If the common difference is 7, how many terms are there in the AP and what is their sum?

Answer:

Here, a = 4, d = 7 and l = 81
Let the n^th term be 81.
Then T_n = 81
⇒ a + (n - 1)d = 4 + (n - 1)7 = 81
⇒ (n - 1)7 = 77
⇒ (n - 1) = 11
⇒ n = 12
Thus, there are 12 terms in the AP.

The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [a + l] ∴ S_{12} = \frac{12}{2} [4 + 81] = 6 \times 85 = 510$

Thus, the required sum is 510.

Question 30:

In an AP the first term is 22, nth term is −11 and sum to first nth terms is 66. Find n and d, the common difference.

Answer:

Here, a = 22, T_n = -11 and S_n = 66
Let d be the common difference of the given AP.
Then T_n = -11
⇒ a + (n - 1)d = 22 + (n - 1)d = -11
⇒ (n - 1)d = -33 ...(i)

The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = 66 \Rightarrow \frac{n}{2} [2 \times 22 + (- 33)] = (\frac{n}{2}) \times 11 = 66 \Rightarrow n = 12$ [Substituting the value of (n - 1)d from (i)]
Putting the value of n in (i), we get:
11d = -33
⇒ d = -3
Thus, n = 12 and d = -3

Question 31:

In an AP the first term is 2, the last term is 29 and sum of the term is 155. Find the common difference of the AP.

Answer:

Here, a = 2, l = 29 and S_n = 155
Let d be the common difference of the given AP and n be the total number of terms.
Then T_n = 29
⇒ a + (n - 1)d = 29
⇒ 2 + (n - 1)d = 29 ...(i)

The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [a + l] = 155 \Rightarrow \frac{n}{2} [2 + 29] = (\frac{n}{2}) \times 31 = 155 \Rightarrow n = 10$
Putting the value of n in (i), we get:
⇒ 2 + 9d = 29
⇒ 9d = 27
⇒ d = 3
Thus, the common difference of the given AP is 3.

Question 32:

Find the sum of the following:
$(1 - \frac{1}{n}) + (1 - \frac{2}{n}) + (1 - \frac{3}{n}) + . . . upto n terms$ .

Answer:

On simplifying the given series, we get:

$(1 - \frac{1}{n}) + (1 - \frac{2}{n}) + (1 - \frac{3}{n}) + . . . n terms = (1 + 1 + 1 + . . . n terms) - (\frac{1}{n} + \frac{2}{n} + \frac{3}{n} + . . . + \frac{n}{n}) = n - (\frac{1}{n} + \frac{2}{n} + \frac{3}{n} + . . . + \frac{n}{n}) Here, (\frac{1}{n} + \frac{2}{n} + \frac{3}{n} + . . . + \frac{n}{n}) is an AP whose first term is \frac{1}{n} and the common difference is (\frac{2}{n} - \frac{1}{n}) = \frac{1}{n} . The sum of n terms of an AP is given by S n = \frac{n}{2} [2 a + (n - 1) d] = n - [\frac{n}{2} \{2 \times (\frac{1}{n}) + (n - 1) \times (\frac{1}{n})\}] = n - [\frac{n}{2} [(\frac{2}{n}) + (\frac{n - 1}{n})]] = n - \{\frac{n}{2} (\frac{n + 1}{n})\} = n - (\frac{n + 1}{2}) = \frac{n - 1}{2}$

Question 33:

The production of TV in a factory increase uniformly by a fixed number every year. It produced 8000 acts in 6th year and 11300 in 9th year. Find the production in (i) first year (ii) 8th year and (iii) 6th year.

Answer:

(i) As the production of TV sets increases uniformly by a fixed number every year, the number of TV sets produced in the 1^st year, 2^nd year, 3^rd year, ... will form an AP. Let the first term be a, common difference of the AP be d and the number of TV sets produced in the n^th year be T_n.

Then T₆ = 8000
⇒ a + 5d = 8000          ...(i)
Also, T₉ = 11300
⇒ a + 8d = 11300          ...(ii)

On subtracting (i) from (ii), we get:
⇒ 3d = 3300
⇒ d = 1100

Putting the value of d in (i), we get:
⇒ a + 5 ⨯ 1100 = 8000
⇒ a = 8000 - 5500 = 2500

Therefore, the production of TV sets in 1^st year is 2500.

(ii) Now, production of TV sets in the 8^th year is given by
T₈ = a + 7d = 2500 + 7 ⨯ 1100 = 10200

(iii) Sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [a + l] \Rightarrow S_{6} = \frac{6}{2} [2500 + 8000] = 3 \times 10500 = 31500$
Thus, the total production of TV sets in the 6^th years is 31500.

Question 34:

A sum of Rs 2800 is to be used to award four prizes. If each prize after the first is Rs 200 less than the preceding prize, find the value of each of the prizes.

Answer:

Let the amounts of the four prizes be Rs a, Rs (a - 200), Rs ( a - 400) and Rs ( a - 600).
Then, a + a - 200 + a - 400 + a - 600 = Rs 2800
⇒ 4a - 1200 = 2800
⇒ 4a = 4000
⇒ a = Rs 1000
Putting the value of a in each term, we get the amounts of the four prizes as follows:
Rs 1000, Rs (1000 - 200) = Rs 800, Rs (1000 - 400) = Rs 600 and Rs (1000 - 600) = Rs 400

Thus, the amounts of prizes are Rs 1000, Rs 800, Rs 600 and Rs 400.

Question 35:

200 logs are stacked in such a way that there are 20 logs in the bottom row, 19 in the next row, 18 in the next row, and so on. In how many rows 200 logs are placed and how many logs are there in the top row?

Answer:

From the bottom, the number of logs in adjacent rows are 20, 19, 18,...
Since the logs decreases uniformly by 1 in each row, the number of logs in the rows form an AP.
Here, a = 20, d = 19 - 20 = -1
Let there be n rows such that S_n = 200.
The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] Here, S_{n} = 200 ∴ S_{n} = 200 \Rightarrow (\frac{n}{2}) \times [2 \times 20 + (n - 1) \times (- 1)] = 200 \Rightarrow (\frac{n}{2}) \times [40 - n + 1] = 200 \Rightarrow n \times [41 - n] = 400 \Rightarrow n^{2} - 41 n + 400 = 0 \Rightarrow n^{2} - 25 n - 16 n + 400 = 0 \Rightarrow n (n - 25) - 16 (n - 25) = 0 \Rightarrow n = 25 o r n = 16$

i.e., the number of rows is either 16 or 25.
For n = 25, we have:
T₂₅ = a + 24 d = 20 + 24 ⨯ (-1) = -4, which is not possible.
Therefore, n = 25 is not acceptable.

For n = 16, we have:
T₁₆ = a +15d = 20 + 15 ⨯ (-1) = 5
Thus, there are 16 rows and 5 logs are placed in the top row.

Question 1:

If $\frac{4}{5}, a,$ 2 are three consecutive terms of an AP, find the value of a.

Answer:

If $\frac{4}{5}$ , a and 2 are three consecutive terms of an AP, then we have:
a - $\frac{4}{5}$ = 2 - a
⇒ 2a = 2 + $\frac{4}{5}$
⇒ 2a = $\frac{14}{5}$
⇒ a = $\frac{7}{5}$

Question 2:

If $(x + 2), 2 x, (2 x + 3)$ are three consecutive terms of an AP, find the value of x.

Answer:

If (x+2),2x and (2x+3) are three consecutive terms of an AP, then we have:
2x - (x+2) = (2x+3) - 2x
⇒ x - 2 = 3
⇒ x = 5
Hence, the value of x is 5.

Question 3:

For what value of p are $(2 p + 1), 13, (5 p - 3)$ three consecutive terms of an AP?

Answer:

Let (2p+1), 13,(5p-3) be three consecutive terms of an AP.
Then 13 - (2p+1) =(5p - 3) - 13
⇒ 7p = 28
⇒ p = 4
∴ When p = 4, (2p+1), 13 and(5p-3) form three consecutive terms of an AP.

Question 4:

For what value of p are $(2 p - 1),$ 7 and 3p three consecutive terms of an AP?

Answer:

Let (2p-1),7 and 3p be three consecutive terms of an AP.
Then 7 - (2p - 1) = 3p - 7
⇒ 5p = 15
⇒ p = 3
∴ When p = 3, (2p -1), 7 and 3p form three consecutive terms of an AP.

Question 5:

The nth term of an AP is (3n + 5). Find its common difference.

Answer:

We have:
T_n = (3n + 5)
Common difference = T₂ - T₁
T₁= 3 ⨯ 1 + 5 = 8
T₂ = 3 ⨯ 2 + 5 = 11
d = 11 - 8 = 3
Hence, the common difference is 3.

Question 6:

The nth term of an AP is (7 − 4n). Find its common difference.

Answer:

We have:
T_n = (7 - 4n)
Common difference = T₂ - T₁
T₁= 7 - 4 ⨯ 1 = 3
T₂ = 7 - 4 ⨯ 2 = -1
d = -1 - 3 = -4
Hence, the common difference is -4.

Question 7:

The first term of an AP is p and its common difference is q. Find its 10th term.

Answer:

Here, a = p and d = q
Now, T_n = a + (n - 1)d
⇒ T_n=p + (n - 1)q
∴ T₁₀ = p + 9q

Question 8:

Write the next term of the AP $\sqrt{8}, \sqrt{18}, \sqrt{32}, . . . . . .$

Answer:

$The given AP is \sqrt{8}, \sqrt{18}, \sqrt{32}, . . . On simplifying the terms, we get : 2 \sqrt{2}, 3 \sqrt{2}, 4 \sqrt{2}, . . . Here, a = 2 \sqrt{2} and d = (3 \sqrt{2} - 2 \sqrt{2}) = \sqrt{2} ∴ Next term, T_{4} = a + 3 d = 2 \sqrt{2} + 3 \sqrt{2} = 5 \sqrt{2} = \sqrt{50}$

Question 9:

Write the next term of the AP $\sqrt{2}, \sqrt{8}, \sqrt{18}, . . . . . .$

Answer:

: $The given AP is \sqrt{2}, \sqrt{8}, \sqrt{18}, . . . On simplifying the terms, we get : \sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2}, . . . Here, a = \sqrt{2} and d = (2 \sqrt{2} - \sqrt{2}) = \sqrt{2} ∴ Next term, T_{4} = a + 3 d = \sqrt{2} + 3 \sqrt{2} = 4 \sqrt{2} = \sqrt{32}$

Question 10:

Which term of the AP 21, 18, 15,... is zero?

Answer:

In the given AP, first term, a = 21 and common difference, d = (18 - 21) = -3
Let's its n^th term be 0.
Then T_n = 0
⇒ a + (n - 1)d = 0
⇒ 21 + (n - 1) ⨯ (-3) = 0
⇒ 24 - 3n = 0
⇒ 3n = 24
⇒ n = 8
Hence, the 8^th term of the given AP is 0.

Question 11:

Which term of the AP 25, 20, 15,... is the first negative term?

Answer:

Here, a = 25 and d = (20 - 25) = -5
Let the n^th term of the given AP be the first negative term.
Then T_n < 0
⇒ a + (n - 1)d < 0
⇒ 25 + (n - 1) ⨯ (- 5) < 0
⇒ 30 - 5n < 0
⇒ 30 < 5n
⇒ n > 6
∴ n = 7

Hence, the 7^th term of the given AP is the first negative term.

Question 12:

Find the 6th term from the end of the AP 5, 7, 9, ...., 201.

Answer:

Here, a = 5, d = (7 - 5) = 2, l = 201 and n = 6^th

Now, n^th term from the end = [l - (n - 1)d]
6^th term from the end = [201 - (6 - 1) ⨯ 2]
= [201 - (5 ⨯ 2)] = (201 - 10) = 191
Hence, the 6^th term from the end of the AP is 191.

Question 13:

Find the sum of first n natural numbers.

Answer:

The first n natural numbers are 1, 2, 3, 4, 5, ..., n.

Here, a = 1 and d = (2 - 1) = 1

Sum of n terms of an AP is given by S_{n} = \frac{n}{2} [2 a + (n - 1) d] = (\frac{n}{2}) \times [2 \times 1 + (n - 1) \times 1] = (\frac{n}{2}) \times [2 + n - 1] = (\frac{n}{2}) \times (n + 1) = \frac{n (n + 1)}{2}

Question 14:

Find the sum of first n even natural numbers.

Answer:

The first n even natural numbers are 2 ,4, 6, 8, 10, ..., n.

Here, a = 2 and d = (4 - 2) = 2

Sum of n terms of an AP is given by S_{n} = \frac{n}{2} [2 a + (n - 1) d] = (\frac{n}{2}) \times [2 \times 2 + (n - 1) \times 2] = (\frac{n}{2}) \times [4 + 2 n - 2] = (\frac{n}{2}) \times (2 n + 2) = n (n + 1)

Hence, the required sum is n(n+1).

Question 15:

Find the sum of the first n odd natural numbers.

Answer:

The first n odd natural numbers are 1, 3, 5, 7, 9, ..., n.

Here, a = 1 and d = (3 - 1) = 2

Sum of n terms of an AP is given by S_{n} = \frac{n}{2} [2 a + (n - 1) d] = (\frac{n}{2}) \times [2 \times 1 + (n - 1) \times 2] = (\frac{n}{2}) \times [2 + 2 n - 2] = (\frac{n}{2}) \times (2 n) = n

²²
Hence, the required sum is n².

Question 1:

If $(x + 2), 2 x, (2 x + 3)$ are three consecutive terms of an AP, then x = ?
(a) 3
(b) 4
(c) 5
(d) 6

Answer:

(c) 5

If (x+2),2x and (2x+3) are three consecutive terms of an AP, then we have:
2x - (x+2) = (2x+3) - 2x
⇒ x - 2 = 3
⇒ x = 5

Question 2:

If $(2 p + 1), 13, (5 p - 3)$ are three consecutive terms of an AP, then p = ?
(a) 4
(b) 5
(c) 6
(d) 3

Answer:

(a) 4
If (2p + 1), 13 and(5p - 3) are three consecutive terms of an AP, then we have:
13 - (2p+1) = (5p-3) - 13
⇒ 12 - 2p = 5p - 16
⇒ 7p = 28
⇒ p = 4

Question 3:

If $(k + 1), 3 k, (4 k + 2)$ are three consecutive terms of an AP, then k = ?
(a) 0
(b) 1
(c) 2
(d) 3

Answer:

(d) 3
If (k+1),3k and(4k+2) are three consecutive terms of an AP, then we have:
3k - (k + 1) = (4k + 2) - 3k
⇒ 2k - 1 = k +2
⇒ k = 3

Question 4:

If $\frac{4}{5}, a, 2$ are three consecutive terms of an AP, then a = ?
(a) 5
(b) $\frac{5}{4}$
(c) $\frac{9}{5}$
(d) $\frac{7}{5}$

Answer:

(d) $\frac{7}{5}$

If $\frac{4}{5}$ , a and 2 are three consecutive terms of an AP, then we have:
a - $\frac{4}{5}$ = 2 - a
⇒ 2a = 2 + $\frac{4}{5}$
⇒ 2a = $\frac{14}{5}$
⇒ a = $\frac{7}{5}$

Question 5:

What is the next term of the AP $\sqrt{2}, \sqrt{8}, \sqrt{18}, . . . ?$
(a) $\sqrt{24}$
(b) $\sqrt{28}$
(c) $\sqrt{30}$
(d) $\sqrt{32}$

Answer:

(d) $\sqrt{32}$ 32−−√
$The given AP is \sqrt{2}, \sqrt{8}, \sqrt{18}, . . . On simplifying the terms, we get : \sqrt{2}, 2 \sqrt{2}, 3 \sqrt{2,} . . . Here, a = \sqrt{2} and d = (2 \sqrt{2} - \sqrt{2}) = \sqrt{2} ∴ Next term, T_{4} = a + 3 d = \sqrt{2} + 3 \sqrt{2} = 4 \sqrt{2} = \sqrt{32}$

Question 6:

What is the next term of the AP $\sqrt{8}, \sqrt{18}, \sqrt{32}, . . . ?$
(a) $\sqrt{40}$
(b) $\sqrt{48}$
(c) $\sqrt{50}$
(d) $\sqrt{54}$

Answer:

(c) $\sqrt{50}$

$The given AP is \sqrt{8}, \sqrt{18}, \sqrt{32}, . . . On simplifying the terms, we get : 2 \sqrt{2}, 3 \sqrt{2}, 4 \sqrt{2}, . . . Here, a = 2 \sqrt{2} and d = (3 \sqrt{2} - 2 \sqrt{2}) = \sqrt{2} ∴ Next term, T_{4} = a + 3 d = 2 \sqrt{2} + 3 \sqrt{2} = 5 \sqrt{2} = \sqrt{50}$

Question 7:

The first term of an AP is p and its common difference is q. What is its 10th term?
(a) p + 10q
(b) q + 10p
(c) p + 9q
(d) q + 9p

Answer:

(c) p + 9q

Here, a = p and d = q
Now, T_n = a + (n - 1)d
⇒ T_n=p + (n - 1)q
⇒ T₁₀ = p + 9q

Question 8:

The number −11, −7, −3, 1, 5, ... are
(a) in AP with d = −18
(b) in AP with d = −14
(c) in AP with d = −4
(d) not in AP

Answer:

(c) in AP with d = 4

In the given progression, [-7 - (-11)] = [-3 - (-7)] = [1 -(-3)] = 4 (constant)
Thus, each term differs from its preceding term by 4.
So, the given progression is an AP with d = 4.

Question 9:

The 30th term of the AP 10, 7, 4, .... is
(a) −87
(b) 87
(c) 77
(d) −77

Answer:

(d) - 77

The given AP is 10, 7, 4, ...
Here, first term, a = 10 and common difference, d = (7 - 10) = -3
Now, T₃₀= a + (30 - 1)d = a + 29d

= 10 + 29 ⨯ (-3) = -77
Hence, 30^th term = -77

Question 10:

The 11th term of the AP −3, $\frac{- 1}{2}, 2, . . . .$ is
(a) 22
(b) 28
(c) −38
(d) $48 \frac{1}{2}$

Answer:

(a) 22
The given AP is -3, $- \frac{1}{2}$ , 2, ...
Here, first term, a = -3 and common difference, d = $[- \frac{1}{2} - (- 3)] = [3 - \frac{1}{2}] = 2 \frac{1}{2}$
$Now, T_{11} = a + (11 - 1) d = a + 10 d = - 3 + 10 \times (2 \frac{1}{2}) = - 3 + (10 \times \frac{5}{2}) = - 3 + 25 = 22$

∴ 11^th term = 22

Question 11:

The first term of an AP is 6 and its common difference is −3. What is the 16th term of the AP?
(a) 51
(b) 39
(c) −39
(d) −42

Answer:

(c) -39

Here, a = 6 and d = -3
Then, T₁₆ = a + 15d = 6 + 15 ⨯ (-3) = -39
Hence, the 16^th term of given AP is -39.

Question 12:

The first two terms of an AP are −5 and 6. Its 21st term is
(a) 115
(b) −115
(c) −215
(d) 215

Answer:

(d) 215
Here, T₁ or a = -5 and T₂ = 6
Then, d = T₂ - T₁= 6 - (-5) = 11
Now, T₂₁ = a + 20d = -5 + 20 ⨯ 11 = 215
Hence, the 21^st term of the given AP is 215.

Question 13:

Which term of the AP 21, 18, 15,...is −81?
(a) 25th
(b) 18th
(c) 35th
(d) 46th

Answer:

(c) 35^th
In the given AP, we have a = 21 and d = (18 - 21) = -3
Let the n^th term be -81.
Then T_n = -81
⇒ a + (n - 1)d = -81
⇒ 21 + (n - 1) ⨯ (-3) = -81
⇒ 24 - 3n = -81
⇒ 3n = 105
⇒ n = 35
Hence, the 35^th terms is -81.

Question 14:

Which term of the AP 72, 63, 54, ... is 0?
(a) 8th
(b) 9th
(c) 10th
(d) 11th

Answer:

(b) 9^th

In the given AP, we have a = 72 and d = (63 - 72) = - 9
Let n^th terms be 0.
Then T_n = 0
⇒ a + (n -1) d = 0
⇒ 72 + (n - 1) ⨯(-9) = 0
⇒ 81 - 9n = 0
⇒ 9n = 81
⇒ n = 9
Hence, the 9^th term is 0.

Question 15:

Which term of the AP 25, 20, 15, ... is the first negative term?
(a) 9th
(b) 8th
(c) 7th
(d) 10th

Answer:

(c) 7^th

Here, a = 25 and d = (20 - 25) = - 5
Let the n^th term of the given AP be the first negative term.
Then T_n_<0
⇒ a + (n -1) d < 0
⇒ 25 + (n - 1) ⨯ (-5) < 0
⇒ 30 - 5n < 0
⇒ 30 < 5n
⇒ 5n > 30
⇒ n > 6
∴ n = 7

Hence, the 7^th term is the first negative term.

Question 16:

If the nth term of an AP is (3n + 5), then its common difference is
(a) 5
(b) 4
(c) 3
(d) 2

Answer:

(c) 3
Here, T_n = 3n + 5
Then, T₁= 3 ⨯ 1 + 5 = 8
Also, T₂= 3 ⨯ 2 + 5 = 11
∴ d = T₂ - T₁ = 11 - 8 = 3

Hence, the common difference is 3.

Question 17:

If the nth term of an AP is (7 − 4n), then its common difference is
(a) 4
(b) −4
(c) 3
(d) −3

Answer:

(b) -4
Here, T_n = 7 - 4n
Then T₁= 7 - 4 ⨯ 1 = 3
Also, T₂= 7 - 4 ⨯ 2 = -1
∴ d = T₂ - T₁= -1 - 3 = -4

Hence, the common difference is -4.

Question 18:

What is the common difference of an AP in which T₁₈ −T₁₄ = 32?
(a) 8
(b) −8
(c) 4
(d) −4

Answer:

(a) 8

The common difference of two terms is constant in an AP.
So, we have:
T₁₈ - T₁₇ = d       ...(i)
T₁₇ - T₁₆ = d       ...(ii)
T₁₆ - T₁₅ = d       ...(iii)
T₁₅ - T₁₄ = d       ...(iv)

On adding the four equations, we get:
T₁₈ - T₁₄ = 4d
However, T₁₈ - T₁₇ = 32
i.e., 4d = 32
⇒ d = 8

Question 19:

In an AP, the 7th term is 4 and the common difference is −4. What is its first term?
(a) 16
(b) 20
(c) 24
(d) 28

Answer:

(d) 28

T₇ = a + 6d
⇒ a + 6 ⨯ (-4) = 4
⇒ a = 4 + 24 = 28
Hence, the first term is 28.

Question 20:

The 4th term of an AP is 14 and its 12th term is 70. What is its first term?
(a) 7
(b) 10
(c) −7
(d) −10

Answer:

( c) -7
T₄ = a + 3d = 14 ...(i)
T₁₂ = a + 11d = 70 ...(ii)
On subtracting (i) from (ii), we get:
⇒ 8d = 56
⇒ d = 7

On putting the value of d in (i), we get:
⇒ a + 3 $\times$ 7 = 14
⇒ a = -7
Hence, the first term is -7.

Question 21:

The 8th term of an AP is 17 and its 14th term is 29. The common difference of the AP is
(a) 3
(b) 2
(c) 5
(d) −2

Answer:

( b) 2
T₈ = a + 7d = 17 ...(i)
T₁₄ = a + 13d = 29 ...(ii)

On subtracting (i) from (ii), we get:
⇒ 6d = 12
⇒ d = 2
Hence, the common difference is 2.

Question 22:

The 17th term of an AP exceeds its 10th term by 21. The common difference of the AP is
(a) 3
(b) 2
(c) −3
(d) −2

Answer:

( a) 3
T₁₀ = a + 9d
T₁₇ = a + 16d

Also, a + 16d = 21 + T₁₀
⇒ a + 16d = 21 + a + 9d
⇒ 7d = 21
⇒ d = 3
Hence, the common difference of the AP is 3.

Question 23:

If the 3rd and 9th terms of an AP are 4 and −8 respectively, then which term of the AP is 0?
(a) 5th
(b) 7th
(c) 11th
(d) 13th

Answer:

( a) 5^th
T₃ = a + 2d = 4 ...(i)
T₉ = a + 8d = -8 ...(ii)
On subtracting (i) from (ii), we get:
⇒ 6d = -12
⇒ d = -2
Putting the value of d in (i), we get:
⇒ a - 4 = 4
⇒ a = 8
Let the T_n term be 0.
Then T_n = 0
⇒ a + (n - 1)d = 0
⇒ 8 + (n - 1) ⨯ (-2) = 0
⇒ 2n = 10
⇒ n = 5
Hence, the 5^th term of the AP is 0.

Question 24:

If 7 times the 7th term of an AP is equal to 11 times the 11th term, then its 18th term will be
(a) 7
(b) 11
(c) 18
(d) 0

Answer:

( d) 0
In the given AP, let the first term be a and the common difference be d.
We have T_n = a + (n - 1)d
Then T₇ = a + (7 - 1)d = a + 6d

Also, T₁₁ = a + (11 - 1)d = a + 10d

Now, (7 ⨯ T₇) = (11 ⨯ T₁₁)
⇒ 7[ a + 6d] = 11[a + 10d]
⇒ 7a + 42d = 11a + 110 d

⇒ 4a = -68d
⇒ a = -17d

∴ T₁₈ = a + (18 - 1)d = a + 17d = -17d + 17d = 0 [ ∵ a = - 17d]
Hence, the 18^th term is 0 (zero).

Question 25:

Which term of the AP 21, 42, 63, 84, ... is 210?
(a) 9th
(b) 10th
(c) 11th
(d) 12th

Answer:

(b) 10^th

Here, a = 21 and d = (42 - 21) = 21
Let 210 be the n^th term of the given AP.
Then T_n = 210
⇒ a + (n - 1)d = 210
⇒ 21 + (n - 1) ⨯ 21= 210
⇒ 21n = 210
⇒ n = 10
Hence, 210 is the 10^th term of the AP.

Question 26:

If 4, x₁, x₂, x₃, 28 are in AP, then x₃ = ?
(a) 19
(b) 23
(c) 22
(d) cannot be determined

Answer:

(c) 22

Here, a = 4, l = 28 and n = 5
Then, T₅ = 28
⇒ a + (n - 1)d = 28
⇒ 4 + (5 - 1)d= 28
⇒ 4d = 24
⇒ d = 6
Hence, x₃ = 28 - 6 = 22

Question 27:

The 20th term from the end of the AP 3, 8, 13, ..., 253 is
(a) 163
(b) 158
(c) 153
(d) 148

Answer:

(b) 158
Here, a = 3, d = (8 - 3) = 5, l = 253 and n = 20^th
Now, n^th term from the end = [l - (n - 1)d]
20^th term from the end = [253 - (20 - 1) ⨯ 5]
= [253 - (19 ⨯ 5)] = (253 - 95) = 158
Hence, the 20^th term from the end of the AP is 158.

Question 28:

The 4th term from the end of AP −11, −8, −5, ... ,49 is
(a) 58
(b) 43
(c) 40
(d) 37

Answer:

(c) 40

Here, a = -11 and d = [-8 - (-11)] = 3, l = 49 and n = 4^th
Now, n^th term from the end = [l - (n - 1)d]
4^th term from the end = [49 - (4 - 1) ⨯ 3]
= [49 - (3 ⨯ 3)] = (49 - 9) = 40
Hence, the 4^th term from the end of the AP is 40.

Question 29:

The second term of an AP is 13 and its 5th term is 25. What is its 17th term.
(a) 73
(b) 77
(c) 69
(d) 81

Answer:

( a) 73

T₂ = a + d = 13 ...(i)
T₅ = a + 4d = 25 ...(ii)
On subtracting (i) from (ii), we get:
⇒ 3d = 12
⇒ d = 4
On putting the value of d in (i), we get:
⇒ a + 4= 13
⇒ a = 9

Now, T₁₇ = a +16d = 9 + 16 ⨯ 4 = 73
Hence, the 17^th term is 73.

Question 30:

The sum of first 16 terms of the AP 10, 6, 2, ..., is
(a) 320
(b) −320
(c) −352
(d) −400

Answer:

(b) - 320

Here, a = 10, d = (6 - 10) = -4 and n = 16
Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{16} = \frac{16}{2} [2 \times 10 + (16 - 1) \times (- 4)] [∵ a = 10, d = - 4 and n = 16] = 8 \times [20 - 60] = 8 \times (- 40) = - 320$
Hence, the sum of the first 16 terms of the given AP is -320.

Question 31:

The sum of first 20 terms of the AP 1, 3, 5, 7, 9, ... is
(a) 264
(b) 400
(c) 472
(d) 563

Answer:

(b) 400

Here, a = 1, d = (3 - 1) = 2 and n = 20
Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{20} = \frac{20}{2} [2 \times 1 + (20 - 1) \times 2] [∵ a = 1, d = 2 and n = 20] = 10 \times [2 + 38] = 10 \times 40 = 400$
Hence, the sum of the first 20 terms of the given AP is 400.

Question 32:

(5 + 13 + 21 + ... + 181) = ?
(a) 2476
(b) 2337
(c) 2219
(d) 2139

Answer:

(d) 2139

Here, a = 5, d = (13-5) = 8 and l = 181
Let the number of terms be n.
Then T_n = 181
         ⇒a + (n-1) d = 181
   ⇒ 5 + ( n -1) ⨯ 8 = 181
   ⇒  8n = 184
   ⇒ n = 23

∴ Required sum =

\frac{n}{2} (a + l)

= \frac{23}{2} (5 + 181) = 23 \times 93 = 2139

Hence, the required sum is 2139.

Question 33:

The sum of n terms of the AP $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, . . . . is$
(a) 1
(b) 2n (n + 1)
(c) $\frac{1}{2} n (n + 1)$
(d) $\frac{1}{\sqrt{2}} n (n + 1)$

Answer:

(d)

\frac{1}{\sqrt{2}} n (n + 1)

H e r e, a = \sqrt{2} and d = \sqrt{8} - \sqrt{2} = 2 \sqrt{2} - \sqrt{2} = \sqrt{2} ∴ S n = \frac{n}{2} [2 a + (n - 1) d] \Rightarrow \frac{n}{2} [2 \times \sqrt{2} + (n - 1) \times \sqrt{2}] \Rightarrow (\frac{n}{2}) (\sqrt{2}) [2 + n - 1] \Rightarrow (\frac{n}{\sqrt{2}}) [n + 1]

Question 34:

How many terms of the AP 3, 7, 11, 15, ... will make the sum 406?
(a) 10
(b) 12
(c) 14
(d) 20

Answer:

(c) 14

Here, a = 3 and d = (7-3) = 4
Let the sum of n terms be 406 .
Then, we have:

S n = \frac{n}{2} [2 a + (n - 1) d] = 406 \Rightarrow \frac{n}{2} [2 \times 3 + (n - 1) \times 4] = 406 \Rightarrow n [3 + 2 n - 2] = 406 \Rightarrow 2 n^{2} + n - 406 = 0 \Rightarrow 2 n^{2} - 28 n + 29 n - 406 = 0 \Rightarrow 2 n (n - 14) + 29 (n - 14) = 0 \Rightarrow (2 n + 29) (n - 14) = 0 \Rightarrow n = 14 (∵ n can' t be a fraction)

Hence, 14 terms will make the sum 406.

Question 35:

The sam of the first 100 natural numbers is
(a) 4950
(b) 5050
(c) 5000
(d) 5150

Answer:

(b) 5050

The first 100 natural numbers are 1, 2, 3, 4, ...,100.

Here, a = 1 and d = (2 - 1) = 1

The sum of n terms of an AP is given by S_{n} = \frac{n}{2} [2 a + (n - 1) d] ∴ S_{100} = (\frac{100}{2}) \times [2 \times 1 + (100 - 1) \times 1] = 50 \times [2 + 99] = 50 \times 101 = 5050

Question 36:

The sum of all odd numbers between 100 and 200 is
(a) 3750
(b) 2352
(c) 2450
(d) 2468

Answer:

(d) 7500

All odd numbers between 100 and 500 are 101, 103, 105, ..., 199.
This is an AP in which a = 101, d = (103 - 101) = 2 and l = 199.

Let the number of terms be n.
Then, T_n = 199
⇒ a + (n-1) d = 199

⇒ 101 + (n - 1) ⨯ 2 = 199
⇒ 2n = 100
⇒ n = 50

∴ Required sum =

\frac{n}{2} (a + l)

= \frac{50}{2} (101 + 199) = 25 \times 300 = 7500

Hence, the required sum is 7500.

Question 37:

The sum of all even natural numbers less than 100 is
(a) 2272
(b) 2352
(c) 2450
(d) 2468

Answer:

(c) 2450

All the even natural numbers less than 100 are 2, 4, 6, 8, ..., 98.
This is an AP in which a = 2, d = (4 - 2) = 2 and l = 98

Let the number of terms be n.
Then T_n = 98
⇒a + (n-1) d = 98

⇒ 2 + ( n - 1) ⨯ 2 = 98
⇒ 2n = 98
⇒ n = 49

∴ Required sum =

\frac{n}{2} (a + l)

= \frac{49}{2} (2 + 98) = 49 \times 50 = 2450

Hence, the required sum is 2450.

Question 38:

The sum of the first fifteen multiples of 8 is
(a) 840
(b) 960
(c) 872
(d) 1081

Answer:

(b) 960

The first 15 multiples of 8 are 8,16,24, 32, ..., 120.
This is an AP in which a = 8, d = (16 - 8) = 8, l = 120 and n = 15

∴ Required sum =

\frac{n}{2} (a + l)

=

(\frac{15}{2}) \times (8 + 120) = 15 \times 64 = 960

Hence, the required sum is 960.

Question 39:

In an AP, the first term is 22, nth term is −11 and the sum of first n terms is 66. The value of n is
(a) 10
(b) 12
(c) 14
(d) 16

Answer:

(b) 12

Here, a = 22, T_n = -11 and S_n = 66
Let d be the common difference of the given AP.
Then T_n = -11
⇒ a + (n - 1)d = 22 + (n-1)d = -11
⇒ (n - 1)d = -33 ...(i)

The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = 66 \Rightarrow \frac{n}{2} [2 \times 22 + (- 33)] = (\frac{n}{2}) \times 11 = 66 \Rightarrow n = 12$ [Substituting the value of (n - 1)d from (i)]

Question 40:

In an AP, the first term is 8, nth term is 33 and the sum of first n terms is 123. Thn, d = ?
(a) 5
(b) −5
(c) 7
(d) 3

Answer:

(a) 5

Here, a = 8, T_n = 33 and S_n = 123
Let d be the common difference of the given AP.
Then, T_n = 33
⇒ a + (n - 1)d = 8 + (n-1)d = 33
⇒ (n - 1)d = 25 ...(i)

The sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = 123 \Rightarrow \frac{n}{2} [2 \times 8 + 25] = (\frac{n}{2}) \times 41 = 123 \Rightarrow n = 6$ [Substituting the value of (n - 1)d from (i)]
Putting the value of n in (i), we get:
5d = 25
⇒ d = 5

Question 41:

The sum of n terms of an AP is given by $S_{n} = (2 n^{2} + 3 n)$ . What is the common difference of the AP?
(a) 3
(b) 4
(c) 5
(d) 9

Answer:

( b) 4
Given: S_n = (2n² + 3n)              ...(i)
Replacing n by (n - 1) in (i), we get:
S_n-1 = 2(n - 1)² + 3(n - 1)
= 2(n² - 2n + 1) + 3n - 3
= 2n² - n - 1
  ∴ T_n = ( S_n - S_n-1)
= (2n² + 3n) - (2n² - n - 1) = 4n + 1
∴ n^th term, T_n = (4n + 1)       ...(ii)

Putting n = 1 in (ii), we get:
T₁ = (4 ⨯ 1) + 1 = 5

Putting n = 2 in (ii), we get:
T₂ = (4 ⨯ 2) + 1 = 9
∴ Common difference, d = T₂ - T₁ = 9 - 5 = 4

Question 42:

The sum of n terms of an AP is given by $S_{n} = (3 n^{2} + 5 n)$ . Which of its terms is 164?
(a) 25th
(b) 26th
(c) 27th
(d) 28th

Answer:

( c) 27^th
Given: S_n = (3n² + 5n)           ...(i)
Replacing n by (n - 1) in (i), we get:
S_n-1 = 3(n - 1)² + 5(n - 1)
= 3(n² - 2n + 1) + 5n - 5
= 3n² - n - 2
  ∴ T_n = (S_n - S_n-1)
= (3n² + 5n) - (3n² - n - 2 ) = 6n + 2
∴ n^th term, T_n = (6n + 2)                ...(ii)

Now, T_n = 164
⇒ (6n + 2) = 164
⇒ 6n = 162
⇒ n = 27
Hence, the 27^th term of AP is 164.

Question 43:

The sum of first p terms of an AP is $(a p^{2} + b p)$ . What is the common difference of the AP?
(a) a
(b) 2a
(c) (a + b)
(d) 3a + b

Answer:

( b) 2a

Given: S_p = (ap² + bp)          ...(i)
Replacing p by (p - 1) in (i), we get:
S_p-1 = a(p - 1)² + b(p - 1)
= a(p² - 2p + 1) + bp - b
= ap² - 2ap + a + bp - b
  ∴ T_p = ( S_p - S_p-1)
          = (ap² + bp) - (ap² - 2ap + a + bp - b )
             = 2ap - a + b
∴ p^th term, T_p = (2ap - a + b)             ...(ii)

Putting p = 1 in (ii), we get:
T₁= (2a ⨯ 1) - a + b = a + b

Putting p = 2 in (ii), we get:
T₂ = (2a ⨯ 2) - a + b = 3a + b
∴ Common difference, d = T₂ - T₁ = 3a + b - a - b = 2a

Question 44:

The famous mathematician associated with finding the sum of first 100 natural numbers, is
(a) Euclid
(b) Gauss
(c) Newton
(d) Pythagoras

Answer:

(b) Gauss is associated with finding the sum of the first 100 natural numbers.

Question 45:

How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?
(a) 71
(b) 72
(c) 73
(d) 74

Answer:

(c) 73

The required numbers are 11, 15, 19, ..., 299.
This is an AP in which a = 11, d = (15 - 11) = 4 and T_n = 299
Now, T_n = 299
⇒ a + (n - 1)d = 299
⇒ 11 + (n - 1) ⨯ 4 = 299
⇒ 4n = 292
⇒ n = 73
Hence, 73 numbers, which when divided by 4 leave a remainder 3, lie in between 10 and 300.

Question 46:

Two APs have the same common difference. Their first terms are −1 and −8, respectively. The difference between their 4th terms is
(a) −1
(b) −8
(c) 7
(d) −9

Answer:

(c) 7

Let the common difference of each AP be d. Then we have:
T₄ = a + (n - 1)d = -1 + 3d
t₄ = a + (n - 1)d = -8 + 3d
∴ (T₄ - t₄) = (-1 + 3d) - (-8 + 3d) = 7
Hence, the difference between their 4^th terms is 7.

Question 47:

The sum of all multiples of 7 between 0 and 500 is
(a) 13916
(b) 17892
(c) 24353
(d) 16984

Answer:

(b) 17892

The multiples of 7 between 0 and 500 are 7, 14, 21, 28, ..., 497.
This is an AP in which a = 7, d = (14 - 7) = 7.
Let the number of terms be n.
Then T_n = 497
⇒ a + (n - 1)d = 497
⇒ 7 + (n - 1) ⨯ 7 = 497
⇒ 7n = 497
⇒ n = 71

∴ S₁₇ = $\frac{n}{2} (a + l) = (\frac{71}{2}) \times (7 + 497) = 71 \times 252 = 17892$

Question 48:

How many natural numbers are there between 23 and 100 which are exactly divisible by 6?
(a) 13
(b) 12
(c) 11
(d) 16

Answer:

(a) 13

The required numbers are 24, 30, 36, 42, ..., 96.
This is an AP in which a = 24, d = (30 - 24) = 6.
Let the number of terms be n.
Then T_n = 96
⇒ a + (n - 1)d = 96
⇒ 24 + (n -1) ⨯ 6 = 96
⇒ 6n = 78
⇒ n = 13

Hence, 13 numbers which are exactly divisible by 6 lie in between 23 and 100.

Question 49:

The sum of all 2-digit numbers divisible by 5 is
(a) 1035
(b) 1245
(c) 1230
(d) 945

Answer:

(d) 945

The 2-digit numbers divisible by 5 are 10, 15, 20, 25, ..., 95.
This is an AP in which a = 10, d = (15 - 10) = 5.
Let the number of terms be n.
Then T_n = 95
⇒ a + (n - 1)d = 95
⇒ 10 + (n - 1) ⨯ 5 = 95
⇒ 5n = 90
⇒ n = 18

∴ S₁₈ = $\frac{n}{2} (a + l) = (\frac{18}{2}) \times (10 + 95) = 9 \times 105 = 945$

Question 50:

Match the following columns

Column I	Column II
(a) How many terms are there in the AP 8, 13, 18, 23,..., 163?	(p) 65
(b) 12th term from the end of the AP 9, 12, 15, 18, ..., 98 is	(q) 25
(c) The sum of the AP 5 + 12 + 19 + 26 + ... + 145 is	(r) 32
(d) The number of terms in the AP (−5) + (−1) + 3 + 7 + ... + 91 is	(s) 1575

Answer:

(a) - (r), (b) -(p), (c) - (s), (d) - (q)

(a) a = 8, d = 5 and l = 163
T_n = a + (n − 1)d = 163
⇒ 8 + (n − 1) ⨯ 5 = 163

⇒ 5n = 160
⇒ n = 32

(b) l = 98 and d = 3
∴ 12^th term from the end of the AP = [l − (n - 1)d]
= 98 - 11 ⨯ 3
= 98 - 33 = 65

(c) Here, a = 5, d = 7 and l = 145
T_n = 5 + (n - 1) ⨯ 7 = 145
⇒ 7n = 147
⇒ n = 21
∴ S₂₁ =

\frac{n}{2} (a + l) = (\frac{21}{2}) \times (5 + 145) = 21 \times 75 = 1575

(d) Here, a = -5, d = 4 and l = 91
T_n = 91
⇒ -5 + (n - 1) ⨯ 4 = 91
⇒ 4n = 100
⇒ n = 25

Question 51:

Match the following columns:

Column I	Column II
(a) In an AP, it is given that T₅ = −9 and T₉ = 7. Then, T₁₄ = ?	(p) 8
(b) In an AP, S_n = (n² + 3n). Then, T₁₆ = ?	(q) 34
(c) In an AP, T₁₀ = 44 and T₁₅ = 64. Then, its first term is	(r) 15
(d) How many 2-digit numbers are divisible by 6?	(s) 27

Answer:

(a) - (s), (b) - (q), (c) - (p ), (d) - ( r)

(a) T₅ = a + 4d = −9      ...(i)
T₉ = a + 8d = 7           ...(ii)
Subtracting (i) from (ii), we get:
4d = 16
∴ d = 4
On putting the value of d in (i), we get a = −25.
∴ T₁₄ = a + 13d = −25 + 13 ⨯ 4 = −25 + 52 = 27

(b)  S_n = (n² + 3n)     ...(i)
Replacing n by (n − 1) in (i), we get;
S_n-1 = (n − 1)² + 3(n − 1)
= (n² − 2n + 1) + 3n − 3
= n² + n − 2
  ∴ T_n = (S_n − S_n-1)
= (n² + 3n) − (n² + n − 2) = 2n + 2
∴ n^th term, T_n = (2n + 2)                ...(ii)
Putting n = 16 in (ii), we get:
T₁₆= (2 ⨯ 16) + 2 = 34
Hence, the 16^th term is 34.

(c) T₁₀ = a + 9d = 44       ...(i)
T₁₅ = a + 14d = 64       ...(ii)
Subtracting (i) from (ii), we get:
5d = 20
∴ d = 4
On putting the value of d in (i), we get a = 8.

(d) All the 2-digit numbers divisible by 6 are 12, 18, 24, 30, ..., 96.
This is an AP whose first term is 12 and the common difference is 6.
Let 96 be the n^th term of the AP.
Now, T_n= 96
⇒ a + (n − 1)d = 96
⇒ 12 + (n -1) 6 = 96
⇒ (n − 1) = 14
⇒ n = 15

Question 52:

Assertion (A)
8th term from the end of the AP 18, 14, 10, 6, ..., −30 is −2.

Reason (R)
The nth term from the end of an AP with last term l and common difference d is l −(n − 1)d.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Answer:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

Explanation:
Here, l = −30, n = 8 and d = −4
∴ 8^th term from the end = l − (n − 1) ⨯ d = −30 − 7 ⨯ (−4) = −30 + 28 = −2
Hence, Assertion (A) is true and clearly, Reason (R) gives Assertion (A).

Question 53:

Assertion (A)
Sum of first n positive integers is given by $S_{n} = \frac{1}{2} n (n + 1) .$
Reason (R)
In an AP with first term a and common difference d, the sum of nth terms is given by $S_{n} = \frac{n}{2} [2 a + (n - 1) d] .$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Answer:

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

$S_{n} = 1 + 2 + 3 + 4 + . . . + n = \frac{n}{2} [2 \times 1 + (n - 1) \times 1] = \frac{n}{2} (n + 1)$
This proves Assertion (A) and clearly, Reason (R) gives Assertion (A).

Question 54:

Assertion (A)
If the first term of an AP is 6, last term is 62 and the sum of the given terms is 510. Then, there are 14 terms in the given AP.

Reason (R)
If a is the first term, l is the last term and n is the number of terms of an AP, then $S_{n} = \frac{n}{2} (a + 1) .$

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R)is true.

Answer:

(d) Assertion (A) is false and Reason (R) is true.

The Reason (R) is clearly true.

$Now, \frac{n}{2} (a + l) = 510 \Rightarrow \frac{n}{2} (6 + 62) = 510 \Rightarrow n = \frac{510}{34} = 15$

Thus, Assertion (A) is false.

Question 1:

Which term of the AP 2, −1, −4, −7, ... is −40?
(a) 8th
(b) 11th
(c) 14th
(d) 15th

Answer:

(d) 15^th

Here, a = 2, d = (−1 − 2) = −3
Let the n^th term be −40.
Then Tn = 40
⇒ a + (n − 1)d = −40
⇒2 + (n − 1) ⨯ (−3) = −40
⇒ 3n = 45
⇒ n = 15

Question 2:

Which term of the AP 20, 17, 14, ... is the first negative term?
(a) 6th
(b) 7th
(c) 8th
(d) 9th

Answer:

( c) 8^th

Here, a = 20 and d = (17 − 20) = −3
Let the n^th term of the given AP be the first negative term.
Then T_n < 0
⇒ a + (n −1)d < 0
⇒ 20 + (n − 1) ⨯ (−3) < 0
⇒ 23 − 3n < 0
⇒ 23 < 3n
⇒ n > 7.6
∴ n = 8
Hence, the 8^th term of the given AP is the first negative term.

Question 3:

What is the sum of all natural numbers from 1 to 100?
(a) 4550
(b) 4780
(c) 5050
(d) 5150

Answer:

(c) 5050

All the natural numbers from 1 to 100 are 1, 2, 3, 4, ..., 100.

Here, a = 1, n = 100 and l = 100
∴ S_n =

\frac{n}{2} (a + l) = \frac{100}{2} (1 + 100) = 50 \times 101 = 5050

Question 4:

If a, (a − 2), 3a are in AP, then a = ?
(a) −3
(b) −2
(c) 2
(d) 3

Answer:

(b) −2

If a, (a − 2) and 3a are in AP, then we have:
(a − 2) − a = 3a − (a − 2)
⇒ 2a = −4
∴ a = −2

Question 5:

How many terms are there in the AP 7, 11, 15, ..., 139?

Answer:

Here, a = 7, d = (11 − 7) = 4 and l = 139
Let there are n terms in the given AP.
Then T_n = 139
⇒ a + (n − 1)d = 139
⇒ 7 + (n − 1) ⨯ 4 = 139
⇒ 4n = 136
⇒ n = 34
Hence, there are 34 terms in the AP.

Question 6:

Is 51 a term of 5, 8, 11, 14, ...,?

Answer:

Here, a = 5, d = (8 − 5) = 3
Let the n^th term of the given AP be 51.
Then T_n = 51
⇒ a + (n − 1)d = 51
⇒ 5 + (n − 1) ⨯ 3 = 51
⇒ 3n = 49
⇒ n = $16 \frac{1}{3}$ , but n cannot be a fraction.
∴ 51 is not a term of the given AP.

Question 7:

Which term of the AP 18, 15, 12,... is 0?

Answer:

In the given AP, first term, a = 18 and common difference, d = (15 − 18) = −3.

Let its n^th term be 0. Then we have:
T_n = 0
⇒ a + (n − 1)d = 0
⇒ 18 + (n − 1) ⨯ (−3) = 0
⇒ 21 − 3n = 0
⇒ 3n = 21
⇒ n = 7
Hence, 0 is the 7^th term of the given AP.

Question 8:

If (5x + 2), (4x − 1) and (x + 2) are in AP, then x = ?

Answer:

Let (5x + 2), (4x − 1) and (x + 2) are consecutive terms of an AP.
Then we have:
(4x − 1) − (5x + 2) = (x + 2) − (4x − 1)
⇒ (−x − 3) = (−3x + 3)
⇒ 2x = 6
⇒ x = 3
∴ If (5x + 2), (4x − 1) and (x + 2) are in AP, then x = 3.

Question 9:

If $S_{n} = (3 n^{2} + 2 n),$ find the first term and the common difference of the AP.

Answer:

Given:
S_n = (3n² + 2n)            ...(i)
Replacing n by (n − 1) in (i), we get:
S_n-1 = 3(n − 1)² + 2(n − 1)
= 3(n² − 2n + 1) + 2n − 2
= 3n² − 4n + 1
  ∴ T_n = ( S_n − S_{n -1})
= (3n² + 2n) − (3n² − 4n + 1) = 6n − 1
∴ n^th term, T_n = (6n − 1)       ...(ii)

Putting n = 1 in (ii), we get:
T₁ = (6 ⨯ 1) − 1 = 5

Putting n = 2 in (ii), we get:
T₂ = (6 ⨯ 2) − 1 = 11
∴ Common difference, d = T₂ − T₁ = 11 − 5 = 6
Thus, a = 5 and d = 6

Question 10:

Find the sum of 12 terms of the AP 5, 8, 11, 14,...

Answer:

Here, a = 5, d = (8 − 5) = 3 and n = 12

Using the formula, $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , we get:
$S_{12} = \frac{12}{2} [2 \times 5 + (12 - 1) \times 3] [∴ a = 5, d = 3 and n = 12] = 6 \times [10 + 33] = 6 \times 43 = 258$

Hence, the sum of the first 12 terms of the given AP is 258.

Question 11:

In an AP, it is given that T₈ = 31 and T₁₅ = 45. Find the AP.

Answer:

Let the first term of the given AP be a and the common difference be d.
Then we have:
T_n = a + (n − 1)d
T₈ = a + 7d = 31 ...(i)
T₁₅ = a + 14d = 45 ...(ii)
Subtracting (i) from (ii), we get:
7d = 14
⇒ d = 2
On putting the value of d in (i), we get:
a = 31 − 14 = 17
∴ The given AP is 17, 19, 21, 23, ...

Question 12:

In an AP, it is given that T_n = (2n + 1). Find the sum of its n terms.

Answer:

We have:
T_n = (2n + 1)
So, T₁= 2 ⨯ 1 + 1 = 3,
T₂= 2 ⨯ 2 + 1 = 5
Now, d = T₂ − T₁
= 5 − 3 = 2
Thus, a = 3 and d = 2
The sum of n terms of an AP is given by

$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [2 \times 3 + (n - 1) \times 2] = n [3 + n - 1] = n (n + 2)$

Question 13:

Find the sum 25 + 28 + 31 + .+ 100.

Answer:

Here, a = 25, d = (28 − 25) = 3 and l = 100
Let there be n terms in the given AP.
Then T_n = 100
     ⇒ a + (n − 1)d = 100
   ⇒ 25 + (n − 1)⨯3 = 100
   ⇒ 3n = 78
   ⇒ n = 26

Sum of n terms of an AP is given by
$S_{n} = \frac{n}{2} [a + l] = \frac{26}{2} [25 + 100] = 13 \times 125 = 1625$

Question 14:

Verify that −130 is a term of the AP 11, 8, 5, 2,.... . Which term is this?

Answer:

Here, a = 11, d = (8 − 11) = −3
Let the n^th term of the given AP be −130.
Then T_n = −130
⇒ a + (n − 1)d = −130
⇒ 11 + (n −1) ⨯ (−3) = −130
⇒ 3n = 144
⇒ n = 48
Thus, −130 is the 48^th term of the given AP.

Question 15:

Find the sum of all 2-digit numbers divisible by 6.

Answer:

The first 2-digit number divisible by 6 is 12 and the largest 2-digit number divisible by 6 is 96.
Thus, the AP is 12, 18, 24, 30, ..., 96, where 96 is the last term of the AP.
Let it be the n^th term of the AP, where a = 12, d = (18 − 12) = 6 and l = 96
Then T_n = 96
⇒ a + (n -1) d = 96
⇒ 12 + (n -1) ⨯ 6 =96
⇒ 6n = 90
⇒n = 15
Hence, there are 15 terms in the AP.

Required sum = $\frac{n}{2} (a + l)$
= $\frac{15}{2} (12 + 96) = 15 \times 54 = 810$
Hence, the sum of all the 2-digit numbers divisible by 6 is 810.

Question 16:

If a = 8, T_n = 63 and S_n = 210, find the value of n.

Answer:

Here, a = 8, T_n = 62 and S_n = 210
Now, a + (n − 1)d = 62
⇒ 8 + (n − 1)d = 62
⇒ (n − 1)d = 54

$Now, S_{n} = \frac{n}{2} [2 a + (n - 1) d] = 210 [As (n - 1) d = 55] \Rightarrow \frac{n}{2} [2 \times 8 + 54] = 210 \Rightarrow 70 n = 210 \times 2 \Rightarrow n = 6$

Question 17:

How many terms of the AP −15, −11, −7, ... are needed to make the sum 744?

Answer:

Here, a = −15 and d = [−11 − (−15)] = 4

Let the required number of terms be n. Then we have:

S_{n} = 744 \Rightarrow \frac{n}{2} [2 a + (n - 1) d] = 744 \Rightarrow \frac{n}{2} [2 \times (- 15) + (n - 1) \times 4] = 744 \Rightarrow \frac{n}{2} [- 30 + 4 n - 4] = 744 \Rightarrow n [2 n - 17] = 744 \Rightarrow 2 n^{2} - 17 n - 744 = 0 \Rightarrow 2 n^{2} - 48 n + 31 n - 744 = 0 \Rightarrow 2 n (n - 24) + 31 (n - 24) = 0 \Rightarrow (2 n + 31) (n - 24) = 0 \Rightarrow n = 24 o r n = - 31 / 2 (n = \frac{- 31}{2} rejected, as n can' t be a fraction)

Hence, 24 terms are needed to make the sum of 744.

Question 18:

Find x, if −4 + (−1) + 2 + ... + x = 437.

Answer:

Here, a = -4 and d = [−1 − (−4)] = 3
Let x be the n^th term of the AP.
Sum of n terms of an AP is given by

S_{n} = \frac{n}{2} [2 a + (n - 1) d] \Rightarrow \frac{n}{2} [2 \times (- 4) + (n - 1) \times 3] = 437 \Rightarrow \frac{n}{2} [- 8 + 3 n - 3] = 437 \Rightarrow n [3 n - 11] = 437 \times 2 \Rightarrow 3 n^{2} - 11 n - 874 = 0 \Rightarrow 3 n^{2} - 57 n + 46 n - 874 = 0 \Rightarrow 3 n (n - 19) + 46 (n - 19) = 0 \Rightarrow (3 n + 46) (n - 19) = 0 \Rightarrow n = 19 o r n = \frac{- 46}{3} (n = \frac{- 46}{3} is rejected, as n can't be a fraction)

Thus, the sum of 19 terms is equal to 437.
∴ x = T₁₉ = a + (n −1)d
⇒ a + 18d
⇒ (−4) + 18 ⨯ 3 = 50

Hence, x = 50

Question 19:

Find the sum of all multiples of 9 lying between 300 and 700.

Answer:

All multiples of 9 lying between 300 and 700 are 306, 315, 324, 333, ..., 693.

This is an AP in which a = 306, d = (315 − 306) = 9 and l = 693.

Let the number of terms be n. Then, we have:
T_n = 693
⇒a + (n − 1) d = 693
⇒ 306 + ( n − 1) ⨯ 9 = 693
⇒ 9n = 396
⇒ n = 44

∴ Required sum =

\frac{n}{2} (a + l)

=

(\frac{44}{2}) \times (306 + 693) = 22 \times 999 = 21978

Hence, the required sum is 21978.

Question 20:

The first and last terms of an AP are 4 and 81 respectively. If the common difference is 7, how many terms are there in the AP and what is thir sum ?

Answer:

Here, a = 4, d = 7 and l = 81

Let its n^thterm be 81.
Then T_n = 81
⇒a + (n-1) d = 81

⇒ 4 + ( n -1) ⨯ 7 = 81
⇒ 7n = 84
⇒ n = 12
Hence, there are 12 terms in the AP.

∴ S_n =

\frac{n}{2} (a + l)

= (\frac{12}{2}) \times (4 + 81) = 6 \times 85 = 510

Hence, the required sum is 510.

Rs Aggarwal 2015 for Class 10 Math Chapter 1 - Real Numbers

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