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#### Question 1:

Find:
(i) 10th term of the A.P. 1, 4, 7, 10, ...
(ii) 18th term of the A.P. ...
(iii) nth term of the A.P. 13, 8, 3, −2, ...

(i) 1, 4, 7, 10...

We have:
a = 1
d =

(ii) ...
We have:

(iii) 13, 8, 3, −2...
We have:

#### Question 2:

If the sequence < an > is an A.P., show that
am +n +amn = 2am.

Let the sequence < an >  be an A.P. with the first term being A and the common difference being D.
To prove: am +n +amn = 2am

LHS: am +n +amn

RHS: 2am

From (i) and (ii), we get:
LHS = RHS
Hence, proved.

#### Question 3:

(i) Which term of the A.P. 3, 8, 13, ... is 248?
(ii) Which term of the A.P. 84, 80, 76, ... is 0?
(iii) Which term of the A.P. 4, 9, 14, ... is 254?

(i) 3, 8, 13...
Here, we have:
a = 3

Hence, 248 is the 50th term of the given A.P.

(ii) 84, 80, 76...
Here, we have:
a = 84

Hence, 0 is the 22nd term of the given A.P.

(iii) 4, 9, 14...
Here, we have:
a = 4

Hence, 254 is the 51st term of the given A.P.

#### Question 4:

(i) Is 68 a term of the A.P. 7, 10, 13, ...?
(ii) Is 302 a term of the A.P. 3, 8, 13, ...?

(i) 7, 10, 13...
Here, we have:
a = 7

Since n is not a natural number.So, 68 is not a term of the given A.P.

(ii) 3, 8, 13...
Here, we have:
a  = 3

Since n is not a natural number.So, 302 is not a term of the given A.P.

#### Question 5:

(i) Which term of the sequence 24, ... is the first negative term?
(ii) Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is (a) purely real (b) purely imaginary?

(i)  24, ...
This is an A.P.
Here, we have:
a = 24

Thus, the 34th term is the first negative term of the given A.P.

(ii)
(a) 12 + 8i, 11 + 6i, 10 + 4i...
This is an A.P.
Here, we have:
a = 12 + 8i

(b) 12 + 8i, 11 + 6i, 10 + 4i...
This is an A.P.
Here, we have:
a = 12 + 8i

#### Question 6:

(i) How many terms are there in the A.P.
7, 10, 13, ... 43?
(ii) How many terms are there in the A.P.

(i) 7, 10, 13...43
Here, we have:
= 7

Let there be n terms in the given A.P.

Thus, there are 13 terms in the given A.P.

(ii)

Here, we have:
= $-1$

Let there be n terms in the given A.P.

Thus, there are 27 terms in the given A.P.

#### Question 7:

The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.

Here, a = 5, d = 3, an = 80
Let the number of terms be n.
Then, we have:

Thus, there are 26 terms in the given A.P.

#### Question 8:

The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.

Given:

We know:

#### Question 9:

If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.

Given:

To prove:

Proof:

LHS = RHS
Hence, proved.

#### Question 10:

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

Given:

To show:

#### Question 11:

The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.

Given:

#### Question 12:

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

Given:

∴ RHS = LHS
Hence, proved.

#### Question 13:

If (m + 1)th term of an A.P. is twice the (n + 1)th term, prove that (3m + 1)th term is twice the (m + n + 1)th term.

Given:

To prove:

$⇒2{a}_{m+n+1}=2\left[\frac{a+3md}{2}\right]\phantom{\rule{0ex}{0ex}}⇒2{a}_{m+n+1}=a+3md$

∴ LHS = RHS

#### Question 14:

If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.

Given:
nth term of the A.P. 9, 7, 5... is the same as the nth term of the A.P. 15, 12, 9...

Equating (i) and (ii), we get:

Thus, 7th terms of both the A.P.s are the same.

#### Question 15:

Find the 12th term from the end of the following arithmetic progressions:
(i) 3, 5, 7, 9, ... 201
(ii) 3, 8, 13, ..., 253
(iii) 1, 4, 7, 10, ..., 88

(i) 3, 5, 7, 9...201
Consider the given progression with 201 as the first term  and −2 as the common difference.
12th term from the end =

(ii) 3, 8, 13...253
Consider the given progression with 253 as the first term  and −5 as the common difference.
12th term from the end =

(iii) 1, 4, 7, 10...88
Consider the given progression with 88 as the first term and −3 as the common difference.
12th term from the end =

#### Question 16:

The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.

Given:
Let the first term of the A.P. be a and the common difference be d.

#### Question 17:

Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.

Given:

#### Question 18:

How many numbers of two digit are divisible by 3?

The two digit numbers that are divisible by 3 are:
12, 15, 18...96, 99
This is an A.P. whose first term is 12 and the common difference is 3.

Thus, there are 30 such terms.

#### Question 19:

An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.

Given:

#### Question 20:

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.

Let a be the first term and d be the common difference. Then,

#### Question 21:

How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?

A number N is divided by 7 leaves a remainder 4.

N = 7q + 4

N can take values 4, 11, 18, ..... 998

Now,
4, 11, 18, ..... 998 are in arithmetic progression.
First term a = 4
common difference d = 7
last term l = 998

We know that,
l = a + (n − 1)d
⇒ 998 = 4 + (n − 1)7
⇒ 998 = 4 + 7n − 7
⇒ 998 = 7n − 3
⇒ 1001 = 7n
$n=\frac{1001}{7}$
n = 143

Hence, 143 numbers are there between 1 and 1000 which when divided by 7 leave remainder 4.

#### Question 22:

The first and the last terms of an A.P. are a and l respectively. Show that the sum of nth term from the beginning and nth term from the end is a + l.

Given:
First term = a
Last term = l
nth term from the beginning  = $a+\left(n-1\right)d$, where d is the common difference.
nth term from the end =
Their sum =

Hence, proved.

#### Question 23:

If < an > is an A.P. such that .

Given:
< an > is an A.P.

#### Question 1:

The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.

#### Question 2:

Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.

#### Question 3:

Find the four numbers in A.P., whose sum is 50 and in which the greatest number is 4 times the least.

#### Question 4:

The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.

#### Question 5:

If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.

#### Question 6:

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

Let the angles be .
Here, = 10
So, are the angles of a quadrilateral whose sum is 360o.

#### Question 1:

Find the sum of the following arithmetic progressions:
(i) 50, 46, 42, ... to 10 terms
(ii) 1, 3, 5, 7, ... to 12 terms
(iii) 3, 9/2, 6, 15/2, ... to 25 terms
(iv) 41, 36, 31, ... to 12 terms
(v) a + b, ab, a − 3b, ... to 22 terms
(vi) (xy)2, (x2 + y2), (x + y)2, ... to n terms
(vii) $\frac{x-y}{x+y},\frac{3x-2y}{x+y},\frac{5x-3y}{x+y}$, ... to n terms

(i) 50, 46, 42 ... to 10 terms

(ii) 1, 3, 5, 7 ... to 12 terms

(iii) 3, 9/2, 6, 15/2 ... to 25 terms

(iv) 41, 36, 31 ... to 12 terms

(v) a + b, ab, a − 3b ... to 22 terms

(vi) (xy)2, (x2 + y2), (x + y)2 ... to n terms

(vii) $\frac{x-y}{x+y},\frac{3x-2y}{x+y},\frac{5x-3y}{x+y}$ ... to n terms

#### Question 2:

Find the sum of the following series:
(i) 2 + 5 + 8 + ... + 182
(ii) 101 + 99 + 97 + ... + 47
(iii) (ab)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]

(i) 2 + 5 + 8 + ... + 182
Here, the series is an A.P. where we have the following:

(ii) 101 + 99 + 97 + ... + 47
Here, the series is an A.P. where we have the following:

(iii) (ab)2 + (a2 + b2) + (a + b)2 + ... + [(a + b)2 + 6ab]
Here, the series is an A.P. where we have the following:

#### Question 3:

Find the sum of first n natural numbers.

The first n natural numbers are:
1, 2, 3, 4...
a = 1, d = 1, Total terms = n

#### Question 4:

Find the sum of all natural numbers between 1 and 100, which are divisible by 2 or 5.

We have to find the sum of all the natural numbers that are divisible by 2 or 5
Required Sum = Sum of the natural numbers between 1 and 100 that are divisible by 2 + Sum of the natural numbers between 1 and 100 that are divisible by 5
− Sum of the natural numbers between 1 and 100 that are divisible by 2 and 5, i.e by 10

$=\left(2+4+6+8+...+98\right)+\left(5+10+15+...+95\right)-\left(10+20+30+...+90\right)\phantom{\rule{0ex}{0ex}}=\frac{50}{2}\left(2+98\right)+\frac{20}{2}\left(5+95\right)-\frac{10}{2}\left(10+90\right)\phantom{\rule{0ex}{0ex}}=2500+1000-500\phantom{\rule{0ex}{0ex}}=3000$

#### Question 5:

Find the sum of first n odd natural numbers.

The first n odd natural numbers are:
1, 3, 5, 7, 9...
a = 1, d = 2, Total terms = n

#### Question 6:

Find the sum of all odd numbers between 100 and 200.

All the odd numbers between 100 and 200 are:
101, 103...199
Here, we have:

$⇒{S}_{50}=7500$

#### Question 7:

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

The odd integers between 1 and 1000 that are divisible by 3 are:
3, 9, 15, 21...999
Here, we have:

#### Question 8:

Find the sum of all integers between 84 and 719, which are multiples of 5.

The integers between 84 and 719, which are multiples of 5 are:
85, 90...715
Here, we have:

#### Question 9:

Find the sum of all integers between 50 and 500 which are divisible by 7.

The integers between 50 and 500 that are divisible by 7 are:
56, 63...497
Here, we have:

#### Question 10:

Find the sum of all even integers between 101 and 999.

The even integers between 101 and 999 are:
102, 104...998
Here, we have:

#### Question 11:

Find the sum of all integers between 100 and 550, which are divisible by 9.

The integers between 100 and 550 that are divisible by 9 are:
108, 117...549
Here, we have:

#### Question 12:

Find the sum of the series:
3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 + ... to 3n terms.

The given sequence i.e., 3 + 5 + 7 + 6 + 9 + 12 + 9 + 13 + 17 +..... to 3n terms.

can be rewritten as 3 + 6 + 9 + .... to n terms + 5 + 9 + 13 + .... to n terms + 7 + 12 + 17 + .... to n terms

Clearly, all these sequence forms an A.P. having n terms with first terms 3, 5, 7 and common difference 3, 4, 5

Hence, required sum = $\frac{n}{2}\left[2×3+\left(n-1\right)3\right]+\frac{n}{2}\left[2×5+\left(n-1\right)4\right]+\frac{n}{2}\left[2×7+\left(n-1\right)5\right]$

$=\frac{n}{2}\left[\left(6+3n-3\right)+\left(10+4n-4\right)+\left(14+5n-5\right)\right]\phantom{\rule{0ex}{0ex}}=\frac{n}{2}\left[12n+18\right]\phantom{\rule{0ex}{0ex}}=3n\left(2n+3\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

#### Question 13:

Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.

The sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7 are:
103, 119...791
Here, we have:
a = 103
d = 16

#### Question 14:

Solve: (i) 25 + 22 + 19 + 16 + ... + x = 115
(ii) 1 + 4 + 7 + 10 + ... + x = 590.

(i) 25 + 22 + 19 + 16 + ... + x = 115
Here, a = 25, d = $-$3,

(ii) 1 + 4 + 7 + 10 + ... + x = 590
Here, a = 1, d = 3,

#### Question 15:

Find the rth term of an A.P., the sum of whose first n terms is 3n2 + 2n.                                                                    [NCERT EXEMPLAR]

#### Question 16:

How many terms are there in the A.P. whose first and fifth terms are −14 and 2 respectively and the sum of the terms is 40?

#### Question 17:

The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.

The progression thus formed is

#### Question 18:

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

#### Question 19:

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

#### Question 20:

The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by , find the number of terms and the series.

Let total number of terms be 2n.
According to question, we have:

#### Question 21:

If Sn = n2 p and Sm = m2 p, mn, in an A.P., prove that Sp = p3.

#### Question 22:

If 12th term of an A.P. is −13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

Let a be the first term and d be the common difference.

#### Question 23:

If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?

#### Question 24:

Find the sum of n terms of the A.P. whose kth terms is 5k + 1.

#### Question 25:

Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.

The two-digit numbers which when divided by 4 yield 1 as remainder are 13, 17....97.

#### Question 26:

If the sum of a certain number of terms of the AP 25, 22, 19, ... is 116. Find the last term.

The given A.P. is 25,.22,.19.....
Here, a = 25, d = $22-25=-3$

#### Question 27:

Find the sum of odd integers from 1 to 2001.

#### Question 28:

How many terms of the A.P. −6, $-\frac{11}{2}$, −5, ... are needed to give the sum −25?

#### Question 29:

In an A.P. the first term is 2 and the sum of the first five terms is one fourth of the next five terms. Show that 20th term is −112.

#### Question 30:

If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, the prove that:
S1 : S2 = (2n + 1) : (n + 1)

#### Question 31:

Find an A.P. in which the sum of any number of terms is always three times the squared number of these terms.

Given:

#### Question 32:

If the sum of n terms of an A.P. is nP + $\frac{1}{2}$ n (n − 1) Q, where P and Q are constants, find the common difference.

#### Question 33:

The sums of n terms of two arithmetic progressions are in the ratio 5n + 4 : 9n + 6. Find the ratio of their 18th terms.

#### Question 34:

The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.

#### Question 1:

If the nth term an of a sequence is given by an = n2n + 1, write down its first five terms.

Thus, the first five terms of the sequence are 1, 3, 7, 13, 21.

#### Question 2:

A sequence is defined by an = n3 − 6n2 + 11n − 6, n Ďµ N. Show that the first three terms of the sequence are zero and all other terms are positive.

Given:
an = n3 − 6n2 + 11n − 6, n Ďµ N

#### Question 3:

Let < an > be a sequence defined by
a1 = 3
and, an = 3an − 1 + 2, for all n > 1
Find the first four terms of the sequence.

Given:
a1 = 3
And, an = 3an − 1 + 2 for all n > 1

#### Question 4:

Let < an > be a sequence. Write the first five terms in each of the following:
(i) a1 = 1, an = an − 1 + 2, n ≥ 2
(ii) a1 = 1 = a2, an = an − 1 + an − 2, n > 2
(iii) a1 = a2 = 2, an = an − 1 − 1, n > 2

(i) a1 = 1, an = an − 1 + 2, n ≥ 2

Hence, the five terms are 1, 3, 5, 7 and 9.

(ii) a1 = 1 = a2, an = an − 1 + an − 2, n > 2

Hence, the five terms are 1, 1, 2, 3 and 5.

(iii) a1 = a2 = 2, an = an − 1 − 1, n > 2

Hence, the five terms are 2, 2, 1, 0 and $-$1.

#### Question 5:

The Fibonacci sequence is defined by
a1 = 1 = a2, an = an − 1 + an − 2 for n > 2

Find $\frac{{}^{a}n+1}{{}^{a}n}$ for n = 1, 2, 3, 4, 5.

a1 = 1 = a2, an = an − 1 + an − 2 for n > 2
Then, we have:

#### Question 6:

Show that each of the following sequences is an A.P. Also find the common difference and write 3 more terms in each case.
(i) 3, −1, −5, −9 ...
(ii) −1, 1/4, 3/2, 11/4, ...
(iii)
(iv) 9, 7, 5, 3, ...

#### Question 7:

The nth term of a sequence is given by an = 2n + 7. Show that it is an A.P. Also, find its 7th term.

#### Question 8:

The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.

We have:
an = 2n2 + n + 1

#### Question 1:

If $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P., prove that:
(i) $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P.
(ii) a (b +c), b (c + a), c (a +b) are in A.P.

#### Question 2:

If a2, b2, c2 are in A.P., prove that $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P.

#### Question 3:

If a, b, c are in A.P., then show that:
(i) a2 (b + c), b2 (c + a), c2 (a + b) are also in A.P.
(ii) b + ca, c + ab, a + bc are in A.P.
(iii) bca2, cab2, abc2 are in A.P.

#### Question 4:

If $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P., prove that:

(i) $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.

(ii) bc, ca, ab are in A.P.

(i) Since $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P., we have:

Hence, $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.

(ii) Since $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P., we have:

Hence, bc, ca, ab are in A.P.

#### Question 5:

If a, b, c are in A.P., prove that:
(i) (ac)2 = 4 (ab) (bc)
(ii) a2 + c2 + 4ac = 2 (ab + bc + ca)
(iii) a3 + c3 + 6abc = 8b3.

Since a, b, c are in A.P., we have:
2b = a+c
$⇒$b = $\frac{a+c}{2}$

(i) Consider RHS:
4 (ab) (bc)

Hence, proved.

(ii) Consider RHS:
2 (ab + bc + ca)

(iii) Consider RHS:
8${b}^{3}$

Hence, proved.

#### Question 6:

If are in A.P., prove that a, b, c are in A.P.

Given:
are in A.P.

#### Question 7:

Show that x2 + xy + y2, z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P., if x, y and z are in A.P.             [NCERT EXEMPLAR]

Hence, x2 + xy + y2z2 + zx + x2 and y2 + yz + z2 are consecutive terms of an A.P.

#### Question 1:

Find the A.M. between:
(i) 7 and 13
(ii) 12 and −8
(iii) (xy) and (x + y).

(i) Let ${A}_{1}$ be the A.M. between 7 and 13.

${A}_{1}$ = $\frac{a+b}{2}$

= $\frac{7+13}{2}$

= 10

(ii) Let ${A}_{1}$ be the A.M. between 12 and −8.

${A}_{1}$ = $\frac{a+b}{2}$
= $\frac{12+\left(-8\right)}{2}$
= 2

(iii) Let ${A}_{1}$ be the A.M. between (xy) and (x + y).
${A}_{1}$ = $\frac{a+b}{2}$
= $\frac{\left(x-y\right)+\left(x+y\right)}{2}$
=  x

#### Question 2:

Insert 4 A.M.s between 4 and 19.

Let be the four A.M.s between 4 and 19. Then, 4, and 19 are in A.P. whose common difference is as follows:
d = $\frac{19-4}{4+1}$
= 3

Hence, the required A.M.s are 7, 10, 13, 16.

#### Question 3:

Insert 7 A.M.s between 2 and 17.

Let be the seven A.M.s between 2 and 17.
Then, 2,   and 17 are in A.P. whose common difference is as follows:

d = $\frac{17-2}{7+1}$

=$\frac{15}{8}$

Hence, the required A.M.s are .

#### Question 4:

Insert six A.M.s between 15 and −13.

Let be the 6 A.M.s between 15 and $-$13.
Then, 15, and $-$13 are in A.P. whose common difference is as follows:

d = $\frac{-13-15}{6+1}$
= $-4$

Hence, the required A.M.s are .

#### Question 5:

There are n A.M.s between 3 and 17. The ratio of the last mean to the first mean is 3 : 1. Find the value of n.

Let be the n arithmetic means between 3 and 17.
Let d be the common difference of the A.P. 3,  and17.
Then, we have:
d = $\frac{17-3}{n+1}$ = $\frac{14}{n+1}$

Now, ${A}_{1}$= 3 + d = 3 + $\frac{14}{n+1}$ = $\frac{3n+17}{n+1}$

And,

#### Question 6:

Insert A.M.s between 7 and 71 in such a way that the 5th A.M. is 27. Find the number of A.M.s.

Let  be the n arithmetic means between 7 and 71.
Thus, there are $\left(n+2\right)$ terms in all.
Let d be the common difference of the above A.P.
Now, a6 = 27
$⇒a+\left(6-1\right)d=27\phantom{\rule{0ex}{0ex}}⇒a+5d=27$
$⇒$d = 4

Also, 71 = an+2
71 = 7+$\left(n+2-1\right)$4
$⇒$ 71 = 7+$\left(n+1\right)$4
$⇒$ n = 15

Therefore, there are 15 A.M.s.

#### Question 7:

If n A.M.s are inserted between two numbers, prove that the sum of the means equidistant from the beginning and the end is constant.

#### Question 8:

If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.

x, y, z are in A.P.
$\therefore$ $y=\frac{x+z}{2}$

Now,
${A}_{1}=\frac{x+y}{2}=\frac{x+\frac{x+z}{2}}{2}=\frac{3x+z}{4}$

And,
${A}_{2}=\frac{y+z}{2}=\frac{\frac{x+z}{2}+z}{2}=\frac{3z+x}{4}$

Let ${A}_{3}$ be the arithmetic mean of .

Hence, proved.

#### Question 9:

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Let be five numbers between 8 and 26.
Let d be the common difference.
Then, we have:
26 = a7
$⇒$26 = 8 + $\left(7-1\right)$d
$⇒$26 = 8 + 6d
$⇒$d = 3

Therefore, the five numbers are 11, 14, 17, 20, 23.

#### Question 6:

A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

It is given that the man counts Rs 180 per minute for half an hour.
∴ Sum of money the man counts in 30 minutes = Rs 180$×$30 = Rs 5400

Total money counted by the man = Rs 10710
∴ Money left for counting after 30 minutes = Rs (10710 − 5400) = Rs 5310

It is given that after 30 minutes, he counts at the rate of Rs 3 less every minute than the preceding minute.

Therefore, it would be an A.P. where a = 177 and d = −3.
Let the time taken to count Rs 5310 be n minutes.

Thus, the time taken to count Rs 5310 would be 59 minutes or 60 minutes.

Hence, the total time taken to count Rs 10710 would be (30 + 59) minutes or (30 + 60) minutes, i.e. 89 minutes or 90 minutes, respectively.

#### Question 1:

A man saved Rs 16500 in ten years. In each year after the first he saved Rs 100 more than he did in the receding year. How much did he save in the first year?

Let the amount saved by the man in the first year be Rs A.
Let d be the common difference.
Let ${S}_{10}$ denote the amount he saves in ten years.
Here, n =10, d =100

We know:

Therefore, the man saved Rs 1200 in the first year.

#### Question 2:

A man saves Rs 32 during the first year. Rs 36 in the second year and in this way he increases his savings by Rs 4 every year. Find in what time his saving will be Rs 200.

Let n be the time in which the man saved Rs 200.
Here, d= 4, a = 32
We know:

Therefore, the man took 5 years to save Rs 200.

#### Question 3:

A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.

Let ${S}_{40}$ denote the total loan amount to be paid in 40 annual instalments.
${S}_{40}$ = 3600
Let Rs a be the value of the first instalment and Rs d be the common difference.
We know:

${S}_{n}=\frac{n}{2}\left\{2a+\left(n-1\right)d\right\}$

On solving equations , we get:
d = 2 and a =51

Hence, the value of the first instalment is Rs 51

#### Question 4:

A manufacturer of radio sets produced 600 units in the third year and 700 units in the seventh year. Assuming that the product increases uniformly by a fixed number every year, find (i) the production in the first year (ii) the total product in 7 years and (iii) the product in the 10th year.

Let ${a}_{n}$ denote the production of radio sets in the nth year.
Here, ${a}_{3}$ = 600, ${a}_{7}$ = 700
We know:
${a}_{n}=a+\left(n-1\right)d$

Solving $\left(1\right)$ and $\left(2\right)$, we get:
d = 25, a = 550
Hence, the production in the first year is 550 units.

(ii)  Let ${S}_{n}$ denote the total production in n years.
Total production in 7 years = ${S}_{7}$

(iii)  Production in the 10th year = ${a}_{10}$

#### Question 5:

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

Let ${S}_{n}$ be the total distance travelled by the gardener.
Let d be the common difference (distance) between two trees. Let a be the distance of the well from the first tree.
Here, n = 25, d = 10, a = 20
Distance travelled by the gardener from the well to the last tree = ${S}_{25}$

Therefore, the total distance the gardener has to travel is 3500 m.

#### Question 7:

A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?

Cost of the piece of equipment at the end of the first year = Rs 600000 – 15% of 600000
= Rs 600000 – Rs 90000
= Rs 510000
Cost of the piece of equipment at the end of the second year = Rs 600000 – 13.5% of 600000
= Rs 600000 – Rs 81000
= Rs 519000
Cost of the piece of equipment at the end of the third year = Rs 600000 – 12% of Rs 600000
= Rs 600000 – Rs 72000
= Rs 528000
The numbers 510000, 519000, 528000 are in an A.P. whose first term in 510000 and common difference is 9000.
∴ Cost of the piece of equipment at the end of 10 years = a + (10 – 1)d
= 510000 + 81000
= Rs 591000

#### Question 8:

A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid amount. How much the tractor cost him?

Cost of the tractor = Rs 12000
It is given that the farmer pays Rs 6000 in cash.
Unpaid amount = Rs 6000
He has to pay Rs 6000 in annual instalments of Rs 500 plus 12% interest on the unpaid amount.
∴ Number of years taken by the farmer to pay the whole amount = 6000 $÷$ 500 = 12
Hence, the interest paid by farmer annually would be as follows:

It is in an A.P. where a = 720 , d = $-$60 and n = 12.
Total sum:

∴ Amount the farmer has to pay = Rs 12000 + Rs 4680 = Rs 16680

#### Question 9:

Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount. How much the scooter will cost him.

Cost of the scooter = Rs 22000
Shamshad Ali pays Rs 4000 in cash.
∴ Unpaid amount = Rs 22000 $-$ Rs 4000 = Rs 18000
Number of years taken by Shamshed Ali to pay the whole amount = 18000 $÷$ 1000 = 18
He agrees to pay the balance in annual instalments of Rs 1000 plus 10% interest on the unpaid amount.

Total amount of instalments:

It is in an A.P. where a = 1800, d = $-$100 and n = 18.
Therefore, total amount of instalments:

∴ Total amount Shamshad Ali has to pay = Rs (22000 + 17100) = Rs 39100

#### Question 10:

The income of a person is Rs 300,000 in the first year and he receives an increase of Rs 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.

Let ${S}_{n}$ denote the total amount the person receives in n years.
Let d be the common increment in his income every year.
Let a denote the initial income of the person.
Here, a = 300,000, d = 10000, n = 20

Total amount at the end of 20 years:

Therefore, the total amount the person receives in 20 years is Rs 79,00,000.

#### Question 11:

A man starts repaying a loan as first instalment of Rs 100 = 00. If he increases the instalments by Rs 5 every month, what amount he will pay in the 30th instalment?

Let ${a}_{30}$ be the amount a man repays in the 30th instalment.
Let d be the common increment in his instalment every month.
Let a be the initial repayment.
Here, a = 100, d = 5, n = 30
Amount to be repaid in the 30th instalment:
${a}_{30}$$⇒$a+$\left(n-1\right)$d
$=100+29×5\phantom{\rule{0ex}{0ex}}=245$

Hence, the man repays Rs 245 in his 30th instalment.

#### Question 12:

A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?                                                                       [NCERT EXEMPLAR]

So, the carpenter takes 12 days to finish the job.

#### Question 13:

We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.                                            [NCERT EXEMPLAR]

We know that,
the sum of the interior angles of a polygon with 3 sides, a1 = 180°,
the sum of the interior angles of a polygon with 4 sides, a2 = 360°,
the sum of the interior angles of a polygon with 5 sides, a3 = 540°,
.
.
.

So, the sum of the interior angles for a 21 sided polygon is 3420°.

#### Question 14:

In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
[NCERT EXEMPLAR]

We have,

the distance travelled to bring the first potato, a1 = 2 $×$ 24 = 48 m,
the distance travelled to bring the second potato, a2 = 2 $×$ (24 + 4) = 56 m,
the distance travelled to bring the third potato, a3 = 2 $×$ (24 + 4 + 4) = 64 m,
.
.
.

So, he would have to run 2480 m to bring back all the potatoes.

#### Question 15:

A man accepts a position with an initial salary of â‚ą5200 per month. It is understood that he will receive an automatic increase of â‚ą320 in the very next month and each month thereafter.
(i) Find his salary for the tenth month.
(ii) What is his total earnings during the first year?

We have,

the initial salary, a1â‚ą5200,
the salary of the second month, a2 = â‚ą5200 + â‚ą320 = â‚ą5520,
the salary of the third month, a3 = â‚ą5520 + â‚ą320 = â‚ą5840,
.
.
.

#### Question 16:

A man saved â‚ą66000 in 20 years. In each succeeding year after the first year he saved â‚ą200 more than what he saved in the previous year. How much did he save in the first year?

As, in each succeeding year after the first year he saved â‚ą200 more than what he saved in the previous year.
So, the savings of each year are in A.P.

We have,
the total savings of the man in 20 years, S20â‚ą66000 and
the difference of his savings in each succeeding year, dâ‚ą200

Let his savings in the first year be a.

Now,

So, he saved â‚ą1400 in the first year.

#### Question 17:

In a cricket team tournament 16 teams participated. A sum of â‚ą8000 is to be awarded among themselves as prize money. If the last place team is awarded â‚ą275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?

We have,
the total sum of prize money to be awarded among 16 teams, S16 = â‚ą8000 and
the prize money awarded to the last place team i.e. a16 = â‚ą275

As, the award increases by the same amount for successive finishing places.
So, the prize money are in A.P.

Let the prize money awarded to the first team be a.

Now,

So, the amount which the first place team will recieve is â‚ą725.

#### Question 1:

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
(a) 87
(b) 88
(c) 89
(d) 90

(c) 89

Solving equations , we get:

a = 4 and d = 5

#### Question 2:

If the sum of p terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
(a) 0
(b) pq
(c) p + q
(d) − (p + q)

(d) $-\left(p+q\right)$

#### Question 3:

If the sum of n terms of an A.P. be 3 n2n and its common difference is 6, then its first term is
(a) 2
(b) 3
(c) 1
(d) 4

(a) 2

#### Question 4:

Sum of all two digit numbers which when divided by 4 yield unity as remainder is
(a) 1200
(b) 1210
(c) 1250
(d) none of these.

(b) 1210

The given series is 13, 17, 21....97.

${a}_{n}=97\phantom{\rule{0ex}{0ex}}⇒a+\left(n-1\right)d=97\phantom{\rule{0ex}{0ex}}⇒13+\left(n-1\right)4=97\phantom{\rule{0ex}{0ex}}⇒n=22\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

Sum of the above series:

#### Question 5:

In n A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is
(a) 6
(b) 8
(c) 4
(d) none of these.

(a) 6
Let be the n arithmetic means between 3 and 17.
Let d be the common difference of the A.P. 3,  and 17.
Then, we have:

d = $\frac{17-3}{n+1}$ = $\frac{14}{n+1}$

Now, ${A}_{1}$= 3 + d = 3 +$\frac{14}{n+1}$ = $\frac{3n+17}{n+1}$

And,

#### Question 6:

If Sn denotes the sum of first n terms of an A.P. < an > such that
(a)

(b)

(c) $\frac{m-1}{n-1}$

(d) $\frac{m+1}{n+1}$

(b)

#### Question 7:

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
(a) 5
(b) 6
(c) 7
(d) 8

(b) 6

#### Question 8:

If the sum of n terms of an A.P., is 3 n2 + 5 n then which of its terms is 164?
(a) 26th
(b) 27th
(c) 28th
(d) none of these.

(b) 27th

#### Question 9:

If the sum of n terms of an A.P. is 2 n2 + 5 n, then its nth term is
(a) 4n − 3
(b) 3 n − 4
(c) 4 n + 3
(d) 3 n + 4

(c) 4 n + 3

#### Question 10:

If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [cosec a1 cosec a2 + cosec a1 cosec a3 + .... + cosec an − 1 cosec an] is
(a) sec a1 − sec an
(b) cosec a1 − cosec an
(c) cot a1 − cot an
(d) tan a1 − tan an

(c) cot a1 − cot an
We have:

#### Question 11:

In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 n terms is
(a) 1/5
(b) 2/3
(c) 3/4
(d) none of these

(a) 1/5

Required ratio: $\frac{{S}_{2n}}{{S}_{4n}-{S}_{2n}}$

#### Question 12:

If a1, a2, a3, .... an are in A.P. with common difference d, then the sum of the series sin d [sec a1 sec a2 + sec a2 sec a3 + .... + sec an − 1 sec an], is
(a) sec a1 − sec an
(b) cosec a1 − cosec an
(c) cot a1 − cot an
(d) tan an − tan a1

(d) tan an − tan a1
We have:

#### Question 13:

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
(a) 5, 10, 15, 20
(b) 4, 10, 16, 22
(c) 3, 7, 11, 15
(d) none of these

(a) 5, 10, 15, 20

Let the four numbers in A.P. be as follows:

Their sum = 50 (Given)

From equations we get:
d  = 5, a = 15

Hence, the numbers are , i.e. 5, 10, 15, 20.

#### Question 14:

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is
(a) 10
(b) 12
(c) 13
(d) 14

(d) 14

The given series is 1, . . . . . . . . . . . , 31
There are n A.M.s between 1 and 31: .

Common difference, d =

Here, we have:
$\frac{{A}_{1}}{{A}_{n}}=\frac{3}{29}\phantom{\rule{0ex}{0ex}}⇒\frac{1+d}{1+nd}=\frac{3}{29}\phantom{\rule{0ex}{0ex}}⇒\frac{1+\frac{30}{n+1}}{1+n×\frac{30}{n+1}}=\frac{3}{29}\phantom{\rule{0ex}{0ex}}⇒\frac{n+1+30}{n+1+30n}=\frac{3}{29}\phantom{\rule{0ex}{0ex}}⇒\frac{n+31}{31n+1}=\frac{3}{29}\phantom{\rule{0ex}{0ex}}⇒29n+899=93n+3\phantom{\rule{0ex}{0ex}}⇒64n=896\phantom{\rule{0ex}{0ex}}⇒n=14$

#### Question 15:

Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn k Sn 1 + Sn − 2 , then k =
(a) 1
(b) 2
(c) 3
(d) none of these

(b) 2

Let the A.P. be a, a+d, a+2d, a+3d...
Given:
$d={S}_{n}-k{S}_{n-1}+{S}_{n-2}$
For n = 3, we have:

$d=\left(3a+3d\right)-k\left(2a+d\right)+a\phantom{\rule{0ex}{0ex}}⇒4a+2d-k\left(2a+d\right)=0\phantom{\rule{0ex}{0ex}}⇒2\left(2a+d\right)=k\left(2a+d\right)\phantom{\rule{0ex}{0ex}}⇒2=k$

#### Question 16:

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by $\frac{{l}^{2}-{a}^{2}}{k-\left(l+a\right)}$, then k =
(a) S
(b) 2S
(c) 3S
(d) none of these

(b) 2S

Given:

$S=\frac{n}{2}\left(l+a\right)\phantom{\rule{0ex}{0ex}}⇒\left(l+a\right)=\frac{2S}{n}$

#### Question 17:

If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =
(a) $\frac{1}{n}$

(b) $\frac{n-1}{n}$

(c) $\frac{n+1}{2n}$

(d) $\frac{n+1}{n}$

(d) $\frac{n+1}{n}$

Given:
Sum of the even natural numbers = k $×$ Sum of the odd natural numbers
$\frac{n}{2}\left\{2a+\left(n-1\right)d\right\}=k×\frac{n}{2}\left\{2a+\left(n-1\right)d\right\}\phantom{\rule{0ex}{0ex}}⇒\left\{2×2+\left(n-1\right)2\right\}=k×\left\{2×1+\left(n-1\right)2\right\}\phantom{\rule{0ex}{0ex}}⇒\frac{4+\left(n-1\right)2}{2+\left(n-1\right)2}=k\phantom{\rule{0ex}{0ex}}⇒\frac{n+1}{n}=k$

#### Question 18:

If the first, second and last term of an A.P are a, b and 2a respectively, then its sum is
(a)

(b) $\frac{ab}{b-a}$

(c)

(d) none of these

(c)

Let the A.P. be a, a+d, a+2d........a+nd.
Here, let d be the common difference and n be the total number of terms.

Given:

From equations we have:
$⇒\frac{a}{n-1}=b-a\phantom{\rule{0ex}{0ex}}⇒\frac{a}{b-a}+1=n\phantom{\rule{0ex}{0ex}}⇒\frac{a+b-a}{b-a}=n\phantom{\rule{0ex}{0ex}}⇒\frac{b}{b-a}=n$

Now, sum of n terms of an A.P.:

#### Question 19:

If, S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then $\frac{{S}_{1}}{{S}_{2}}$ =
(a) $\frac{2n}{n+1}$

(b) $\frac{n}{n+1}$

(c) $\frac{n+1}{2n}$

(d) $\frac{n+1}{n}$

(a) $\frac{2n}{n+1}$

Let n be an odd number.
Given:

From equations , we have:

#### Question 20:

If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to
(a) $\frac{1}{2}{p}^{3}$

(b) mn p

(c) P3

(d) (m + n) p2

(c) p3

Given:

From , we have:

Substituting d = 2p in equation $\left(1\right)$, we get:
a = p

Sum of p terms of the A.P. is given by:

#### Question 21:

Mark the correct alternative in the following question:

If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is

(a) $-$p                         (b) p                         (c) + q                         (d) p $-$ q

Hence, the correct alternative is option (b).

#### Question 22:

Mark the correct alternative in the following question:

The 10th common term between the A.P.s 3, 7, 11, 15, ... and 1, 6, 11, 16, ... is

(a) 191                         (b) 193                         (c) 211                         (d) none of these

Hence, the correct alternative is option (a).

#### Question 23:

Mark the correct alternative in the following question:

Hence, the correct alternative is option (c).

#### Question 24:

Mark the correct alternative in the following question:

Let Sn denote the sum of first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to

(a) 4                                  (b) 6                                  (c) 8                                  (d) 10

Hence, the correct alternative is option (b).

#### Question 25:

If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is
(a) 3
(b) 2
(c) 6
(d) 4

Let Sn denote sum of n terms of an A.P
Such that Sn = 3n + 2n2
i.e S1 = 3(1) + 2(1)2
S1 = 3 + 2 = 5
where S1 = a (first term only)
S2 = a + (a + d) where a + d represents second term and d is common difference.
$\begin{array}{rcl}{S}_{2}& =& 3\left(2\right)+2{\left(2\right)}^{2}\\ & =& 6+2\left(4\right)\\ & =& 6+8\\ {S}_{2}& =& 14\end{array}$
i.e 2a + d = 14
i.e 10 + d = 14
d = 4
i.e common difference of A.P is 4

Hence, the correct answer is option D.

#### Question 26:

If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is
(a) 0
(b) 22
(c) 220
(d) 198

Let a denote the first term of A.P and d denote the common difference of the A.P
Then ninth term is a + 8d and 13th term is a + 12d
According to given condition,
9(a + 8d) = 13(a + 12d)
i.e 9a + 72d = 13a + 156d
i.e –156d + 72d = 4a
i.e –84d = 4a
i.e a = – 21d
i.e a + 21d = 0
which represents 22nd term of A.P.
∴ The 22nd term of the A.P is 0
Hence, the correct answer is option A.

#### Question 1:

The sum of the terms equidistant from the beginning and end in an A.P. is always same and is equal to the sum of ________ and ________ terms.

Let a1, a2, ............ an be an A.P with common difference d.
Then kth term is ak = a1 + (k – 1)d and kth term from last is the (n – k + 1)th term from beginning

kth term from beginning + kth term from the end is
${a}_{1}+\left(k-1\right)d+{a}_{1}+\left(n-k\right)d\phantom{\rule{0ex}{0ex}}=2{a}_{1}+d\left(k-1+n-k\right)d\phantom{\rule{0ex}{0ex}}=2{a}_{1}+\left(n-1\right)d\phantom{\rule{0ex}{0ex}}={a}_{1}+\left({a}_{1}+\left(n-1\right)d\right)\phantom{\rule{0ex}{0ex}}={a}_{1}+{a}_{n}$
∴ Sum of the terms equidistant from the beginning of an A.P and end of A.P is always same and is equal to sum of first term and nth term or last term of the A.P.

#### Question 2:

The minimum value of 4x + 41 – x, x ∈ R, is ___________.

Since
Arithmetic mean ≥ geometric mean of 4x and 41 – x

i.e minimum value of 4x + 41 – x is 4.

#### Question 3:

If the first, second and last terms of an A.P. are a, b and 2a respectively, then the sum of its terms is ___________.

Let first term of A.P be a
Second term of A.P be b and last term of A.P be given by 2a
Since d = a2a1 = ba

∴ Sn
i.e sum of n terms of A.P is

Sum of its term = $\frac{3ab}{2\left(b-a\right)}$

#### Question 4:

The number of terms in an A.P. whose first term is 10, last term is 50 and the sum of all terms is 300, is ___________.

First term of A.P is given  = 10
Last term of A.P is given = 50
Sum of all terms Sn = 300        (given)
Since sum of all terms

Hence, number of terms of an A.P is 10.

#### Question 5:

The arithmetic mean of first n natural numbers is ___________.

First n natural numbers are 1, 2, 3, ...........n
Sum of these numbers is 1 + 2 + 3 + ........... + n $=\frac{n\left(n+1\right)}{2}$

i.e Arithmetic mean of first n natural number = $\frac{n+1}{2}$

#### Question 6:

The sum of first n odd natural numbers is ___________.

First n odd natural numbers are 1, 3, 5, 7, ............. n
Here first term a = 1
Common difference d = 2
Sum of A.P is

i.e Sum of first n even numbers = n(n + 1).

#### Question 7:

The sum of first n even natural numbers is ___________.

Ans

#### Question 8:

If n is even, then the sum of first n terms of the series 1 – 2 + 3 – 4 + 5 – 6 +..., is ___________.

For even number n,
Consider 1 – 2 + 3 – 4 + 5 – 6 + ................
for 1 – 2 = – 1
3 – 4 = – 1
5 – 6 = –1
i.e Combining two terms at a time given –1
So, if in all, n even terms are there
Sum of the expression 1 – 2 + 3 – 4 + .......... n – 1 – n.
$=-\frac{n}{2}$

#### Question 9:

If twice the 11th term of an A.P is equal to 7 times of its 21st terms, then the value of 25th term is ___________.

Let us suppose a denote the first term of A.P and d represents the common difference
Then 11th term is a + 10d
21st term is a + 20d
According to given condition,
2(11th term) = 7(21st term)
i.e 2(a + 10d) = 7(a + 20d)
i.e 2a + 20d = 7a + 140d
i. 5a + 120d = 0
i.e 5(a + 24d) = 0
Since 5 ≠ 0
a + 24d = 0
i.e 25th term of A.P is 0.

#### Question 10:

If the sums of n terms of two arithmetic progressions are the ratio (2n + 3) : (6n + 5), then the ratio of their 13th terms is ___________.

Let a1 and d2 represent the first term and common difference of first arithmetic progression.
Also, a2 and d2 represent the first term  and common difference of second arithmetic progression then sum of n terms of first A.P
${\left({S}_{n}\right)}_{1}=\frac{n}{2}\left\{2{a}_{1}+\left(n-1\right){d}_{1}\right\}$
and sum of n terms of second A.P
${\left({S}_{n}\right)}_{2}=\frac{n}{2}\left\{2{a}_{2}+\left(n-1\right){d}_{2}\right\}$
Now, according to given condition

Since nth term of any A.P is a + (n – 1)d
∴ 13th term of first A.P is a1 + 12d1 and 13th term of second A.P is a2 + 12d2

i.e ratio of their 13th terms is $\frac{53}{155}$.

#### Question 11:

The sum of n arithmetic means between a and b is ___________.

Let a and b be two given numbers
Let A1, A2, ______ An be arithmetic means between a and b.
a, A1, A2, _____ An , b are in A.P
Here total number of terms is n + 2 first term is a last term is b.
∴ Sum of first n + 2 terms = $\frac{\left(n+2\right)}{2}\left\{a+b\right\}$

i.e

#### Question 12:

If the sum of n arithmetic means between 9 and 51 is 270, then the value of n is ___________.

Let a = 9
b = 51
Given sum of n arithmetic means between 9 and 51 is 270.
Let A1, A2, ______ An be arithmetic means between a and b.
a, A1, A2, _____ An , b are in A.P
Here total number of terms is n + 2 first term is a last term is b.
∴ Sum of first n + 2 terms = $\frac{\left(n+2\right)}{2}\left\{a+b\right\}$

Hence, value of n is 9.

#### Question 13:

The sum of 4 arithmetic means between 3 and 23 is ___________.

Let a = 3
b = 23
To find sum of 4 arithmetic term ..............
here n = 4
∴ Sum of 4 arithmetic terms is means between 3 and 23

i.e sum of 4 arithmetic means between 3 and 23 = 52.

#### Question 14:

If $\frac{{a}^{n+1}+{b}^{n+1}}{{a}^{n}+{b}^{n}}$ is the A.M. of a and b, then n = ___________.

Given:- $\frac{{a}^{n+1}+{b}^{n+1}}{{a}^{n}+{b}^{n}}$ is arithmetic mean of a and b.
Since arithmetic mean between a and b is $\frac{a+b}{2}$

#### Question 1:

Write the common difference of an A.P. whose nth term is xn + y.

Common difference of an A.P., d = ${a}_{2}-{a}_{1}$

#### Question 2:

Write the common difference of an A.P. the sum of whose first n terms is $\frac{p}{2}{n}^{2}+Qn$.

Sum of the first n terms of an A.P. = $\frac{p}{2}{n}^{2}+Qn$
Sum of one term of an A.P. = ${S}_{1}$
$⇒\frac{p}{2}{\left(1\right)}^{2}+Q\left(1\right)\phantom{\rule{0ex}{0ex}}⇒\frac{p}{2}+Q$

Sum of two terms of an A.P. = ${S}_{2}$
$⇒\frac{p}{2}{\left(2\right)}^{2}+Q\left(2\right)\phantom{\rule{0ex}{0ex}}⇒2p+2Q$
Now, we have:

Common difference:

#### Question 3:

If the sum of n terms of an AP is 2n2 + 3n, then write its nth term.

Given:
${S}_{n}=2{n}^{2}+3n$

Common difference, d  = ${a}_{2}-{a}_{1}$
= 9 $-$ 5
= 4
nth term = a +$\left(n-1\right)d$
= 5+$\left(n-1\right)$4
= 4n+1

#### Question 4:

If log 2, log (2x − 1) and log (2x + 3) are in A.P., write the value of x.

The numbers log 2, log (2x − 1) and log (2x + 3) are in A.P.

Disclaimer: The question in the book has some error, so, this solution is not matching with the solution given in the book. This solution here is created according to the question given in the book.

#### Question 5:

If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their mth terms.

Given:

Ratio of their m terms = $\frac{{a}_{m}}{{b}_{m}}$
To find the ratio of the mth terms, replace n by 2m$-$1 in equation (1).
$⇒\frac{2a+\left(2m-2\right)d}{2b+\left(2m-2\right){d}^{1}}=\frac{2\left(2m-1\right)+5}{3\left(2m-1\right)+4}\phantom{\rule{0ex}{0ex}}⇒\frac{a+\left(m-1\right)d}{b+\left(m-1\right){d}^{1}}=\frac{4m-2+3}{6m-3+4}\phantom{\rule{0ex}{0ex}}⇒\frac{{a}_{m}}{{b}_{m}}=\frac{4m+1}{6m+1}\phantom{\rule{0ex}{0ex}}$

#### Question 6:

Write the sum of first n odd natural numbers.

We need to find the sum of 1, 3, 5, 7... upto n terms.
Here, a = 1, d = 2

We know:

Therefore, the sum of the first n odd numbers is ${n}^{2}$.

#### Question 7:

Write the sum of first n even natural numbers.

We need to find the sum of 2, 4, 6, 8...upto n terms.
Here, a = 2, d = 2

We know:

Therefore, the sum of the first n odd numbers is $n\left(n+1\right)$.

#### Question 8:

Write the value of n for which nth terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.

For the first series, a = 3, ${d}_{1}$= 7
For the second series, b = 63, ${d}_{2}$= 2

Given:
${a}_{n}={b}_{n}\phantom{\rule{0ex}{0ex}}⇒a+\left(n-1\right){d}_{1}=b+\left(n-1\right){d}_{2}\phantom{\rule{0ex}{0ex}}⇒3+\left(n-1\right)7=63+\left(n-1\right)2\phantom{\rule{0ex}{0ex}}⇒3+7n-7=63+2n-2\phantom{\rule{0ex}{0ex}}⇒5n=65\phantom{\rule{0ex}{0ex}}⇒n=13$

Hence, the 13th terms of both the series are the same.

#### Question 9:

If 7, then find the value of n.

#### Question 10:

If mth term of an A.P. is n and nth term is m, then write its pth term.

Given:

Solving equations , we get:
d = $-1$
a = n+m$-1$

pth term:

Hence, the pth term is n + m $-$p.

#### Question 11:

If the sums of n terms of two AP.'s are in the ratio (3n + 2) : (2n + 3), then find the ratio of their 12th terms.