Rd Sharma Xi 2018 Solutions for Class 12 Science Math Chapter 20 Geometric Progressions are provided here with simple step-by-step explanations. These solutions for Geometric Progressions are extremely popular among Class 12 Science students for Math Geometric Progressions Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma Xi 2018 Book of Class 12 Science Math Chapter 20 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma Xi 2018 Solutions. All Rd Sharma Xi 2018 Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.

#### Question 2:

Show that the sequence <an>, defined by an = $\frac{2}{{3}^{n}}$, n ϵ N is a G.P.

#### Question 3:

Find:
(i) the ninth term of the G.P. 1, 4, 16, 64, ...

(ii) the 10th term of the G.P.

(iii) the 8th term of the G.P. 0.3, 0.06, 0.012, ...

(iv) the 12th term of the G.P.

(v) nth term of the G.P.

(vi) the 10th term of the G.P.

#### Question 4:

Find the 4th term from the end of the G.P.

#### Question 5:

Which term of the progression 0.004, 0.02, 0.1, ... is 12.5?

#### Question 6:

Which term of the G.P. :

(i)

(ii)

(iii)

(iv)

#### Question 7:

Which term of the progression 18, −12, 8, ... is $\frac{512}{729}$?

#### Question 8:

Find the 4th term from the end of the G.P.

#### Question 9:

The fourth term of a G.P. is 27 and the 7th term is 729, find the G.P.

#### Question 10:

The seventh term of a G.P. is 8 times the fourth term and 5th term is 48. Find the G.P.

#### Question 11:

If the G.P.'s 5, 10, 20, ... and 1280, 640, 320, ... have their nth terms equal, find the value of n.

#### Question 12:

If 5th, 8th and 11th terms of a G.P. are p. q and s respectively, prove that q2 = ps.

#### Question 13:

The 4th term of a G.P. is square of its second term, and the first term is − 3. Find its 7th term.

#### Question 14:

In a GP the 3rd term is 24 and the 6th term is 192. Find the 10th term.

#### Question 15:

If a, b, c, d and p are different real numbers such that:
(a2 + b2 + c2) p2 − 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d are in G.P.

#### Question 16:

If $\frac{a+bx}{a-bx}=\frac{b+cx}{b-cx}=\frac{c+dx}{c-dx}$ (x ≠ 0), then show that a, b, c and d are in G.P.

#### Question 17:

If the pth and qth terms of a G.P. are q and p, respectively, then show that (p + q)th term is ${\left(\frac{{q}^{p}}{{p}^{q}}\right)}^{\frac{1}{p-q}}$.

#### Question 1:

Find three numbers in G.P. whose sum is 65 and whose product is 3375.

Let the terms of the the given G.P. be
Then, product of the G.P. = 3375
$⇒$ a3 = 3375
$⇒$a = 15
Similarly, sum of the G.P. = 65

Substituting the value of a

Hence, the G.P. for a = 15 and r = $\frac{1}{3}$ is 45, 15, 5.
And, the G.P. for a = 15 and r = 3 is 5, 15, 45.

#### Question 2:

Find three numbers in G.P. whose sum is 38 and their product is 1728.

Let the terms of the the given G.P. be
Then, product of the G.P. = 1728
$⇒$ a3 = 1728
$⇒$a = 12
Similarly, sum of the G.P. = 38

Substituting the value of a

Hence, the G.P. for a = 12 and r$\frac{2}{3}$ is 18, 12 and 8.
And, the G.P. for  a = 12 and r = $\frac{3}{2}$ is 8, 12 and 18.

Hence, the three numbers are 8, 12 and 18.

#### Question 3:

The sum of first three terms of a G.P. is 13/12 and their product is − 1. Find the G.P.

Let the first three numbers of the given G.P. be .
∴ Product of the G.P. = −1
$⇒$ a3 = −1
$⇒$a = −1
Similarly, Sum of the G.P. = $\frac{13}{12}$

Substituting the value of a = −1

Hence, the G.P. for a = −1 and r = $-\frac{3}{4}$ is .

And, the G.P. for a = −1 and r = $-\frac{4}{3}$ is .

#### Question 4:

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $871}{2}$. Find them.

Let the required numbers be
Product of the G.P. = 125

Sum of the products in pairs = $87\frac{1}{2}=\frac{175}{2}$

Hence, the G.P. for a = 5 and r = $\frac{1}{2}$ is 10, 5 and $\frac{5}{2}$.
And, the G.P. for a = 5 and r = 2 is $\frac{5}{2}$, 5 and 10.

#### Question 5:

The sum of first three terms of a G.P. is $\frac{39}{10}$ and their product is 1. Find the common ratio and the terms.

Let the terms of the G.P be .
∴ Product of the G.P. = 1
$⇒{a}^{3}=1\phantom{\rule{0ex}{0ex}}⇒a=1$
Now, sum of the G.P. = $\frac{39}{10}$

Hence, putting the values of a and r , the required numbers are .

#### Question 6:

The sum of three numbers in G.P. is 14. If the first two terms are each increased by 1 and the third term decreased by 1, the resulting numbers are in A.P. Find the numbers.

Let the numbers be a, ar and ar2.

According to the question, a + 1, ar + 1 and ar2 − 1 are  in A.P.

Similarly putting r = 2 in (ii), we get a = 2.
So, the G.P is 2, 4 and 8.

#### Question 7:

The product of three numbers in G.P. is 216. If 2, 8, 6 be added to them, the results are in A.P. Find the numbers.

Let the terms of the given G.P. be .
∴ Product = 216
$⇒{a}^{3}=216\phantom{\rule{0ex}{0ex}}⇒a=6$
It is given that are in A.P.

#### Question 8:

Find three numbers in G.P. whose product is 729 and the sum of their products in pairs is 819.

Let the required numbers be
Product of the G.P. = 729
$⇒{a}^{3}=729\phantom{\rule{0ex}{0ex}}⇒a=9$
Sum of the products in pairs = 819

#### Question 9:

The sum of three numbers in G.P. is 21 and the sum of their squares is 189. Find the numbers.

Let the required numbers be
Sum of the numbers = 21

Sum of the squares of the numbers = 189

#### Question 1:

Find the sum of the following geometric progressions:
(i) 2, 6, 18, ... to 7 terms;
(ii) 1, 3, 9, 27, ... to 8 terms;
(iii) 1, −1/2, 1/4, −1/8, ... to 9 terms;
(iv) (a2b2), (ab), $\left(\frac{a-b}{a+b}\right)$, ... to n terms;
(v) 4, 2, 1, 1/2 ... to 10 terms.

(i) Here, a = 2 and r = 3.

(ii) Here, a = 1 and r = 3.

(iii) Here, a = 1 and r = −$\frac{1}{2}$.

(iv) Here, a = a2b2 and r = $\frac{1}{a+b}$.

(v) Here, a = 4 and r = $\frac{1}{2}$

#### Question 2:

Find the sum of the following geometric series:
(i) 0.15 + 0.015 + 0.0015 + ... to 8 terms;

(ii)

(iii)

(iv) (x +y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + ... to n terms;

(v)

(vi) $\frac{a}{1+i}+\frac{a}{\left(1+i{\right)}^{2}}+\frac{a}{\left(1+i{\right)}^{3}}+...+\frac{a}{\left(1+i{\right)}^{n}}.$

(vii) 1, −a, a2, −a3, ... to n terms (a ≠ 1)

(viii) x3, x5, x7, ... to n terms

(ix)

(i) Here, a = 0.15 and r $=\frac{{a}_{2}}{{a}_{1}}=\frac{0.015}{0.15}=\frac{1}{10}$.

(ii) Here, a = $\sqrt{2}$ and r = $\frac{1}{2}$.
${\mathrm{S}}_{8}=\mathrm{a}\left(\frac{1-{\mathrm{r}}^{8}}{1-\mathrm{r}}\right)\phantom{\rule{0ex}{0ex}}=\sqrt{2}\left(\frac{1-{\left(\frac{1}{2}\right)}^{8}}{1-\frac{1}{2}}\right)\phantom{\rule{0ex}{0ex}}=\sqrt{2}\left(\frac{1-\frac{1}{256}}{\frac{1}{2}}\right)\phantom{\rule{0ex}{0ex}}=2\sqrt{2}\left(\frac{255}{256}\right)\phantom{\rule{0ex}{0ex}}=\frac{255\sqrt{2}}{128}$

(iii) Here, a =.

#### Question 3:

Evaluate the following:
(i) $\sum _{n=1}^{11}\left(2+{3}^{n}\right)$

(ii) $\sum _{k=1}^{n}\left({2}^{k}+{3}^{k-1}\right)$

(iii) $\sum _{n=2}^{10}{4}^{n}$

(i)

(ii)

(iii)

#### Question 4:

Find the sum of the following series:
(i) 5 + 55 + 555 + ... to n terms;
(ii) 7 + 77 + 777 + ... to n terms;
(iii) 9 + 99 + 999 + ... to n terms;
(iv) 0.5 + 0.55 + 0.555 + ... to n terms.
(v) 0.6 + 0.66 + 0.666 + .... to n terms

(i) We have,
5 + 55 + 555+ ... n terms
Taking 5 as common:
${S}_{n}$ = 5[1 + 11 + 111 + ... n terms]

(ii) We have,
7 + 77 + 777 + ... n terms
${\mathrm{S}}_{n}$= 7 [1 + 11 + 111 + ... n terms]

(iii) We have,
9 + 99 + 999 + ... n terms

(iv) We have,
0.5 + 0.55 + 0.555 + ... n terms
${S}_{n}$= 5 [0.1 + 0.11+0.111 + ... n terms]

(v) We have,
0.6 + 0.66 +.666 + ... to n terms
${S}_{n}$= 6 [0.1 + 0.11+ 0.111 + ... n terms]

#### Question 5:

How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make $\frac{3069}{512}$?

Here, a = 3
Common ratio,
Sn = $\frac{3069}{512}$

#### Question 6:

How many terms of the series 2 + 6 + 18 + ... must be taken to make the sum equal to 728?

Here, a = 2
Common ratio, r = 3
Sum of n terms, Sn = 728

#### Question 7:

How many terms of the sequence ... must be taken to make the sum $39+13\sqrt{3}$?

Here, a = $\sqrt{3}$
Common ratio, r$\sqrt{3}$
Sum of n terms, Sn = $39+3\sqrt{3}$

#### Question 8:

The sum of n terms of the G.P. 3, 6, 12, ... is 381. Find the value of n.

Here, a = 3
Common ratio, r = 3
Sum of n terms, Sn = 381
∴ Sn = 3 + 6 + 12 + ... + n terms

#### Question 9:

The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.

Here, common ratio, r = 3
nth term, an = 486
Sn = 728

#### Question 10:

The ratio of the sum of first three terms is to that of first 6 terms of a G.P. is 125 : 152. Find the common ratio.

Let a be the first term and r be the common ratio of the G.P.

#### Question 11:

The 4th and 7th terms of a G.P. are respectively. Find the sum of n terms of the G.P.

Let a be the first term and r be the common ratio of the G.P.

Find the sum :

#### Question 13:

The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.

Let a be the first term and r be the common ratio of the G.P.

#### Question 14:

If S1, S2, S3 be respectively the sums of n, 2n, 3n terms of a G.P., then prove that ${S}_{1}^{2}+{S}_{2}^{2}$ = S1 (S2 + S3).

Let a be the first term and r be the common ratio of the given G.P.

#### Question 15:

Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from (n + 1)th to (2n)th term is $\frac{1}{{r}^{n}}$.

Let a be the first term and r be the common ratio of the G.P.
Sum of the first n terms of the series =
Similarly,

#### Question 16:

If a and b are the roots of x2 − 3x + p = 0 and c, d are the roots x2 − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (qp) = 17 : 15.

We have,
a +b = 3, ab = p, c + d =12 and cd = q
a, b, c and d form a G.P.
∴ First term = ab = ar, c = ar2 and d = ar3
Then, we have
a + b = 3  and c + d = 12

#### Question 17:

How many terms of the G.P. 3, $\frac{3}{2},\frac{3}{4}$..... are needed to give the sum $\frac{3069}{512}$?

#### Question 18:

A person has 2 parents, 4 grandparents, 8 great grandparents, and so on. Find the number of his ancestors during the ten generations preceding his own.

Here, the ancestors of the person form the G.P.  2, 4, 8, 16, ........
Now, first term, a = 2
And,  r = 2
∴ Number of  his ancestors during the ten generations preceding his own,

#### Question 19:

If S1, S2, ..., Sn are the sums of n terms of n G.P.'s whose first term is 1 in each and common ratios are 1, 2, 3, ..., n respectively, then prove that S1 + S2 + 2S3 + 3S4 + ... (n − 1) Sn = 1n + 2n + 3n + ... + nn.

#### Question 20:

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.

Let there be 2n terms in the given G.P. with the first term being a and the common ratio being r.
According to the question
Sum of all the terms = 5 (Sum of the terms occupying the odd places)

#### Question 21:

Let an be the nth term of the G.P. of positive numbers. Let such that α ≠ β. Prove that the common ratio of the G.P. is α/β.

Let a be the first term and r be the common ratio of the G.P.

#### Question 22:

Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.

#### Question 1:

Find the sum of the following series to infinity:

(i) $1-\frac{1}{3}+\frac{1}{{3}^{2}}-\frac{1}{{3}^{3}}+\frac{1}{{3}^{4}}+...\infty$

(ii) 8 + $4\sqrt{2}$ + 4 + ... ∞

(iii) 2/5 + 3/52 +2/53 + 3/54 + ... ∞.

(iv) 10 − 9 + 8.1 − 7.29 + ... ∞

(v) $\frac{1}{3}+\frac{1}{{5}^{2}}+\frac{1}{{3}^{3}}+\frac{1}{{5}^{4}}+\frac{1}{{3}^{5}}+\frac{1}{56}+...\infty$

#### Question 2:

Prove that: (91/3 . 91/9 . 91/27 ... ∞) = 3.

#### Question 3:

Prove that: (21/4 . 41/8 . 81/16. 161/32 ... ∞) = 2.

#### Question 4:

If Sp denotes the sum of the series 1 + rp + r2p + ... to ∞ and sp the sum of the series 1 − rp + r2p − ... to ∞, prove that Sp + sp = 2 . S2p.

#### Question 5:

Find the sum of the terms of an infinite decreasing G.P. in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81.

#### Question 6:

Express the recurring decimal 0.125125125 ... as a rational number.

#### Question 7:

Find the rational number whose decimal expansion is $0.4\overline{)23}$.

#### Question 8:

Find the rational numbers having the following decimal expansions:
(i) $0.\overline{)3}$
(ii) $0.\overline{)231}$
(iii) $3.5\overline{)2}$
(iv) $0.6\overline{)8}$

#### Question 9:

One side of an equilateral triangle is 18 cm. The mid-points of its sides are joined to form another triangle whose mid-points, in turn, are joined to form still another triangle. The process is continued indefinitely. Find the sum of the (i) perimeters of all the triangles. (ii) areas of all triangles.

According to the midpoint theorem, the sides of each triangle formed by joining the midpoints of an equilateral triangle are half of the sides of the equilateral triangle. In other words, the triangles formed are equilateral triangles with sides 18 cm, 9 cm, 4.5 cm, 2.25 cm, ...

#### Question 10:

Find an infinite G.P. whose first term is 1 and each term is the sum of all the terms which follow it.

Here, first term, a = 1
Common ratio = r

#### Question 11:

The sum of first two terms of an infinite G.P. is 5 and each term is three times the sum of the succeeding terms. Find the G.P.

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

#### Question 12:

Show that in an infinite G.P. with common ratio r (|r| < 1), each term bears a constant ratio to the sum of all terms that follow it.

#### Question 13:

If S denotes the sum of an infinite G.P. S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively .

#### Question 1:

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

a ,b and c are in G.P.

#### Question 2:

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

a, b, c are in G.P.

#### Question 3:

Find k such that k + 9, k − 6 and 4 form three consecutive terms of a G.P.

k, k + 9, k−6 are in G.P.

But, k = 0 is not possible.
k = 16

#### Question 4:

Three numbers are in A.P. and their sum is 15. If 1, 3, 9 be added to them respectively, they form a G.P. Find the numbers.

Let the first term of an A.P. be a and its common difference be d.

#### Question 5:

The sum of three numbers which are consecutive terms of an A.P. is 21. If the second number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a G.P. Find the numbers.

Let the first term of an A.P is a and its common difference be d.

#### Question 6:

The sum of three numbers a, b, c in A.P. is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a G.P. Find a, b, c.

Let the first term of the A.P. be a and the common difference be d.
a = a , b = a + d and c = a + 2d

#### Question 7:

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an A.P. Find the numbers.

Let the first term of a G.P be a and its common ratio be r.

#### Question 8:

If a, b, c are in G.P., prove that:

(i) a (b2 + c2) = c (a2 + b2)

(ii) ${a}^{2}{b}^{2}{c}^{2}\left(\frac{1}{{a}^{3}}+\frac{1}{{b}^{3}}+\frac{1}{{c}^{3}}\right)={a}^{3}+{b}^{3}+{c}^{3}$

(iii) $\frac{\left(a+b+c{\right)}^{2}}{{a}^{2}+{b}^{2}+{c}^{2}}=\frac{a+b+c}{a-b+c}$

(iv) $\frac{1}{{a}^{2}-{b}^{2}}+\frac{1}{{b}^{2}}=\frac{1}{{b}^{2}-{c}^{2}}$

(v) (a + 2b + 2c) (a − 2b + 2c) = a2 + 4c2.

a, b and c are in G.P.
.......(1)

#### Question 9:

If a, b, c, d are in G.P., prove that:

(i) $\frac{ab-cd}{{b}^{2}-{c}^{2}}=\frac{a+c}{b}$

(ii) (a + b + c + d)2 = (a + b)2 + 2 (b + c)2 + (c + d)2

(iii) (b + c) (b + d) = (c + a) (c + d)

a, b, c and d are in G.P.

.......(1)

#### Question 10:

If a, b, c are in G.P., prove that the following are also in G.P.:
(i) a2, b2, c2
(ii) a3, b3, c3
(iii) a2 + b2, ab + bc, b2 + c2

a, b and c are in G.P.

#### Question 11:

If a, b, c, d are in G.P., prove that:

(i) (a2 + b2), (b2 + c2), (c2 + d2) are in G.P.

(ii) (a2b2), (b2c2), (c2d2) are in G.P.

(iii)

(iv) (a2 + b2 + c2), (ab + bc + cd), (b2 + c2 + d2) are in G.P.

a, b, c and d are in G.P.

.......(1)

#### Question 12:

If (a b), (bc), (ca) are in G.P., then prove that (a + b + c)2 = 3 (ab + bc + ca)

#### Question 13:

If a, b, c are in G.P., then prove that: $\frac{{a}^{2}+ab+{b}^{2}}{bc+ca+ab}=\frac{b+a}{c+b}$

#### Question 14:

If the 4th, 10th and 16th terms of a G.P. are x, y and z respectively. Prove that x, y, z are in G.P.

#### Question 15:

If a, b, c are in A.P. and a, b, d are in G.P., then prove that a, ab, dc are in G.P.

#### Question 16:

If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that pq, qr, rs are in G.P.

#### Question 17:

If $\frac{1}{a+b},\frac{1}{2b},\frac{1}{b+c}$ are three consecutive terms of an A.P., prove that a, b, c are the three consecutive terms of a G.P.

Here,

#### Question 18:

If xa = xb/2 zb/2 = zc, then prove that $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.

#### Question 19:

If a, b, c are in A.P., b,c,d are in G.P. and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P., prove that a, c,e are in G.P.

#### Question 20:

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.

#### Question 21:

If a, b, c are in A.P. and a, b, d are in G.P., show that a, (ab), (dc) are in G.P.

#### Question 22:

If a, b, c are three distinct real numbers in G.P. and a + b + c = xb, then prove that either x < −1 or x > 3.

#### Question 23:

If pth, qth and rth terms of an A.P. and G.P. are both a, b and c respectively, show that ${a}^{b-c}{b}^{c-a}{c}^{a-b}=1$.

Let A be the first term and D be the common difference of the AP. Therefore,

Also, suppose A' be the first term and R be the common ratio of the GP. Therefore,

Now,

Subtracting (2) from (1), we get

Subtracting (3) from (2), we get

Subtracting (1) from (3), we get

$\therefore {a}^{b-c}{b}^{c-a}{c}^{a-b}$

$={\left[A\text{'}{R}^{\left(p-1\right)}\right]}^{\left(q-r\right)D}×{\left[A\text{'}{R}^{\left(q-1\right)}\right]}^{\left(r-p\right)D}×{\left[A\text{'}{R}^{\left(r-1\right)}\right]}^{\left(p-q\right)D}$                 [Using (4), (5), (6), (7), (8) and (9)]

$=A{\text{'}}^{\left(q-r\right)D}{R}^{\left(p-1\right)\left(q-r\right)D}×A{\text{'}}^{\left(r-p\right)D}{R}^{\left(q-1\right)\left(r-p\right)D}×A{\text{'}}^{\left(p-q\right)D}{R}^{\left(r-1\right)\left(p-q\right)D}$

$=A{\text{'}}^{\left[\left(q-r\right)D+\left(r-p\right)D+\left(p-q\right)D\right]}×{R}^{\left[\left(p-1\right)\left(q-r\right)D+\left(q-1\right)\left(r-p\right)D+\left(r-1\right)\left(p-q\right)D\right]}$

$=A{\text{'}}^{\left[q-r+r-p+p-q\right]D}×{R}^{\left[pq-pr-q+r+qr-pq-r+p+pr-qr-p+q\right]D}\phantom{\rule{0ex}{0ex}}={\left(A\text{'}\right)}^{0}×{R}^{0}\phantom{\rule{0ex}{0ex}}=1×1\phantom{\rule{0ex}{0ex}}=1$

#### Question 1:

Insert 6 geometric means between 27 and $\frac{1}{81}$.

#### Question 2:

Insert 5 geometric means between 16 and $\frac{1}{4}$.

#### Question 3:

Insert 5 geometric means between $\frac{32}{9}\mathrm{and}\frac{81}{2}$.

#### Question 4:

Find the geometric means of the following pairs of numbers:
(i) 2 and 8
(ii) a3b and ab3
(iii) −8 and −2

#### Question 5:

If a is the G.M. of 2 and $\frac{1}{4}$, find a.

#### Question 6:

Find the two numbers whose A.M. is 25 and GM is 20.

#### Question 7:

Construct a quadratic in x such that A.M. of its roots is A and G.M. is G.

#### Question 8:

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio .

#### Question 9:

If AM and GM of roots of a quadratic equation are 8 and 5 respectively, then obtain the quadratic equation.

#### Question 10:

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

#### Question 11:

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.

#### Question 12:

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that:

#### Question 13:

If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that $\frac{{G}_{1}^{2}}{{G}_{2}}+\frac{{G}_{2}^{2}}{{G}_{1}}=2A$.

#### Question 1:

If the fifth term of a G.P. is 2, then write the product of its 9 terms.

Here, a5 = 2

Product of the nine terms, i.e. :

#### Question 2:

If (p + q)th and (pq)th terms of a G.P. are m and n respectively, then write is pth term.

#### Question 3:

If logxa, ax/2 and logb x are in G.P., then write the value of x.

#### Question 4:

If the sum of an infinite decreasing G.P. is 3 and the sum of the squares of its term is $\frac{9}{2}$, then write its first term and common difference.

Let us take a G.P. whose first term is and common difference is r.

#### Question 5:

If pth, qth and rth terms of a G.P. are x, y, z respectively, then write the value of xqr yrp zpq.

Let us take a G.P. whose first term is A and common ratio is R.

#### Question 6:

If A1, A2 be two AM's and G1, G2 be two GM's between a and b, then find the value of .

#### Question 7:

If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.

#### Question 8:

Write the quadratic equation the arithmetic and geometric means of whose roots are A and G respectively.

#### Question 9:

Write the product of n geometric means between two numbers a and b.

#### Question 10:

If a = 1 + b + b2 + b3 + ... to ∞, then write b in terms of a.

#### Question 1:

If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is
(a) 1/10
(b) 1/11
(c) 1/9.
(d) 1/20

(b) $\frac{1}{11}$

Let the first term of the G.P. be a.
Let its common ratio be r.
​According to the question, we have:
First term = 10        [Sum of all successive terms]

#### Question 2:

If the first term of a G.P. a1, a2, a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is
(a) −2/5
(b) −3/5
(c) 2/5
(d) none of these

(a) −$\frac{2}{5}$

If the first term is 1, then, the G.P. will be

#### Question 3:

If a, b, c are in A.P. and x, y, z are in G.P., then the value of xbc yca zab is
(a) 0
(b) 1
(c) xyz
(d) xa yb zc

(b) 1

#### Question 4:

The first three of four given numbers are in G.P. and their last three are in A.P. with common difference 6. If first and fourth numbers are equal, then the first number is
(a) 2
(b) 4
(c) 6
(d) 8

(d) 8

#### Question 5:

If a, b, c are in G.P. and a1/x = b1/y = c1/z, then xyz are in
(a) AP
(b) GP
(c) HP
(d) none of these

(a) AP

#### Question 6:

If S be the sum, P the product and R be the sum of the reciprocals of n terms of a GP, then P2 is equal to
(a) S/R
(b) R/S
(c) (R/S)n
(d) (S/R)n

(d) ${\left(\frac{S}{R}\right)}^{n}$

#### Question 7:

The fractional value of 2.357 is
(a) 2355/1001
(b) 2379/997
(c) 2355/999
(d) none of these

(c)$\frac{2355}{999}$

#### Question 8:

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

(a) $\frac{p-q}{q-r}$

(b) $\frac{q-r}{p-q}$

(c) pqr

(d) none of these

(b) $\frac{q-r}{p-q}$

Let a be the first term and d be the common difference of the given A.P.
Then, we have:

#### Question 9:

The value of 91/3 . 91/9 . 91/27 ... upto inf, is
(a) 1
(b) 3
(c) 9
(d) none of these

#### Question 10:

The sum of an infinite G.P. is 4 and the sum of the cubes of its terms is 92. The common ratio of the original G.P. is
(a) 1/2
(b) 2/3
(c) 1/3
(d) −1/2

(a) 1/2

Disclaimer: None of the given options are correct. This solution has been created according to the question given in the book.

#### Question 11:

If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
(a) 1/3
(b) 2/3
(c) 1/4
(d) 3/4

(d) 3/4

#### Question 12:

The nth term of a G.P. is 128 and the sum of its n terms  is 225. If its common ratio is 2, then its first term is
(a) 1
(b) 3
(c) 8
(d) none of these

Disclaimer: None of the given options are correct. This solution has been created according to the question given in the book.

#### Question 13:

If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is
(a) 1/4
(b) 1/2
(c) 2
(d) 4

(d) 4

#### Question 14:

If a, b, c are in G.P. and x, y are AM's between a, b and b,c respectively, then

(a) $\frac{1}{x}+\frac{1}{y}=2$

(b) $\frac{1}{x}+\frac{1}{y}=\frac{1}{2}$

(c) $\frac{1}{x}+\frac{1}{y}=\frac{2}{a}$

(d) $\frac{1}{x}+\frac{1}{y}=\frac{2}{b}$

(d) $\frac{1}{x}+\frac{1}{y}=\frac{2}{b}$

#### Question 15:

If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to
(a) $\frac{{p}^{3}+{q}^{3}}{pq}$

(b) $\frac{{p}^{3}-{q}^{3}}{pq}$

(c) $\frac{{p}^{2}+{q}^{2}}{2}$

(d) $\frac{pq}{2}$

(a) $\frac{{p}^{3}+{q}^{3}}{pq}$

#### Question 16:

If p, q be two A.M.'s and G be one G.M. between two numbers, then G2 =
(a) (2p q) (p − 2q)
(b) (2pq) (2qp)
(c) (2pq) (p + 2q)
(d) none of these

(a) (2p q) (p − 2q)

#### Question 17:

If x is positive, the sum to infinity of the series

(a) 1/2
(b) 3/4
(c) 1
(d) none of these

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

#### Question 18:

If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is
(a) 7
(b) 8
(c) 9
(d) 10

(b) 8

#### Question 19:

Given that x > 0, the sum $\sum _{n=1}^{\infty }{\left(\frac{x}{x+1}\right)}^{n-1}$ equals
(a) x

(b) x + 1

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

(d) $\frac{x+1}{2x+1}$

(b) x + 1

#### Question 20:

In a G.P. of even number of terms, the sum of all terms is five times the sum of the odd terms. The common ratio of the G.P. is
(a) $-\frac{4}{5}$
(b) $\frac{1}{5}$
(c) 4
(d) none of these

(c) 4

#### Question 21:

Let x be the A.M. and y, z be two G.M.s between two positive numbers. Then, $\frac{{y}^{3}+{z}^{3}}{xyz}$ is equal
to
(a) 1
(b) 2
(c) $\frac{1}{2}$
(d) none of these

(b) 2

#### Question 22:

The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to
(a) 64
(b) 16
(c) 32
(d) 0

(a) 64

#### Question 23:

The two geometric means between the numbers 1 and 64 are
(a) 1 and 64
(b) 4 and 16
(c) 2 and 16
(d) 8 and 16
(e) 3 and 16

(b) 4 and 16

#### Question 24:

In a G.P. if the (m + n)th term is p and (mn)th term is q, then its mth term is
(a) 0
(b) pq
(c) $\sqrt{pq}$
(d) $\frac{1}{2}\left(p+q\right)$

(c) $\sqrt{pq}$

#### Question 25:

Mark the correct alternative in the following question:

Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then p2R3 : S3 is equal to

(a) 1 : 1                        (b) (Common ratio)n : 1                        (c) (First term)2 : (Common ratio)2                        (d) None of these

Hence, the correct alternative is option (a).

#### Question 1:

Show that each one of the following progressions is a G.P. Also, find the common ratio in each case:

(i) 4, −2, 1, −1/2, ...

(ii) −2/3, −6, −54, ...

(iii)

(iv) 1/2, 1/3, 2/9, 4/27, ...