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# RD Sharma solutions for Class 11 Mathematics chapter 20 - Geometric Progression

## Chapter 20: Geometric Progression

Ex. 20.10Ex. 20.20Ex. 20.30Ex. 20.40Ex. 20.50Ex. 20.60Others

#### Chapter 20: Geometric Progression Exercise 20.10 solutions [Pages 9 - 10]

Ex. 20.10 | Q 1.1 | Page 9

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

4, −2, 1, −1/2, ...

Ex. 20.10 | Q 1.2 | Page 9

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

−2/3, −6, −54, ...

Ex. 20.10 | Q 1.3 | Page 9

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

$a, \frac{3 a^2}{4}, \frac{9 a^3}{16}, . . .$

Ex. 20.10 | Q 1.4 | Page 9

Show that one of the following progression is a G.P. Also, find the common ratio in case:1/2, 1/3, 2/9, 4/27, ...

Ex. 20.10 | Q 2 | Page 10

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

Ex. 20.10 | Q 3.1 | Page 10

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

Ex. 20.10 | Q 3.2 | Page 10

Find:

the 10th term of the G.P.

$- \frac{3}{4}, \frac{1}{2}, - \frac{1}{3}, \frac{2}{9}, . . .$

Ex. 20.10 | Q 3.3 | Page 10

Find :

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

Ex. 20.10 | Q 3.4 | Page 10

Find :

the 12th term of the G.P.

$\frac{1}{a^3 x^3}, ax, a^5 x^5 , . . .$

Ex. 20.10 | Q 3.5 | Page 10

Find :

nth term of the G.P.

$\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .$

Ex. 20.10 | Q 3.6 | Page 10

Find :

the 10th term of the G.P.

$\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .$

Ex. 20.10 | Q 4 | Page 10

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

$\frac{2}{27}, \frac{2}{9}, \frac{2}{3}, . . . , 162$
Ex. 20.10 | Q 5 | Page 10

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

Ex. 20.10 | Q 6.1 | Page 10

Which term of the G.P. :

$\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, \frac{1}{4\sqrt{2}}, . . . \text { is }\frac{1}{512\sqrt{2}}?$

Ex. 20.10 | Q 6.2 | Page 10

Which term of the G.P. :

$2, 2\sqrt{2}, 4, . . .\text { is }128 ?$

Ex. 20.10 | Q 6.3 | Page 10

Which term of the G.P. :

$\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?$

Ex. 20.10 | Q 6.4 | Page 10

Which term of the G.P. :

$\frac{1}{3}, \frac{1}{9}, \frac{1}{27} . . \text { . is } \frac{1}{19683} ?$

Ex. 20.10 | Q 7 | Page 10

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

Ex. 20.10 | Q 8 | Page 10

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

$\frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \frac{1}{54}, . . . , \frac{1}{4374}$

Ex. 20.10 | Q 9 | Page 10

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

Ex. 20.10 | Q 10 | Page 10

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

Ex. 20.10 | Q 11 | Page 10

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

Ex. 20.10 | Q 12 | Page 10

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

Ex. 20.10 | Q 13 | Page 10

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

Ex. 20.10 | Q 14 | Page 10

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

Ex. 20.10 | Q 15 | Page 10

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.

Ex. 20.10 | Q 16 | Page 10

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

Ex. 20.10 | Q 17 | Page 10

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

#### Chapter 20: Geometric Progression Exercise 20.20 solutions [Page 16]

Ex. 20.20 | Q 1 | Page 16

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

Ex. 20.20 | Q 2 | Page 16

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

Ex. 20.20 | Q 3 | Page 16

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

Ex. 20.20 | Q 4 | Page 16

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87\frac{1}{2}$ . Find them.

Ex. 20.20 | Q 5 | Page 16

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.

Ex. 20.20 | Q 6 | Page 16

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.

Ex. 20.20 | Q 7 | Page 16

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.

Ex. 20.20 | Q 8 | Page 16

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

Ex. 20.20 | Q 9 | Page 16

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

#### Chapter 20: Geometric Progression Exercise 20.30 solutions [Pages 27 - 29]

Ex. 20.30 | Q 1.1 | Page 27

Find the sum of the following geometric progression:

2, 6, 18, ... to 7 terms;

Ex. 20.30 | Q 1.2 | Page 27

Find the sum of the following geometric progression:

1, 3, 9, 27, ... to 8 terms;

Ex. 20.30 | Q 1.3 | Page 27

Find the sum of the following geometric progression:

1, −1/2, 1/4, −1/8, ... to 9 terms;

Ex. 20.30 | Q 1.4 | Page 27

Find the sum of the following geometric progression:

(a2 − b2), (a − b), $\left( \frac{a - b}{a + b} \right)$ to n terms;

Ex. 20.30 | Q 1.5 | Page 27

Find the sum of the following geometric progression:

4, 2, 1, 1/2 ... to 10 terms.

Ex. 20.30 | Q 2.1 | Page 27

Find the sum of the following geometric series:

0.15 + 0.015 + 0.0015 + ... to 8 terms;

Ex. 20.30 | Q 2.2 | Page 27

Find the sum of the following geometric series:

$\sqrt{2} + \frac{1}{\sqrt{2}} + \frac{1}{2\sqrt{2}} + . . .\text { to 8 terms };$

Ex. 20.30 | Q 2.3 | Page 27

Find the sum of the following geometric series:

$\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };$

Ex. 20.30 | Q 2.4 | Page 27

Find the sum of the following geometric series:

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

Ex. 20.30 | Q 2.5 | Page 27

Find the sum of the following geometric series:

$\frac{3}{5} + \frac{4}{5^2} + \frac{3}{5^3} + \frac{4}{5^4} + . . . \text { to 2n terms };$

Ex. 20.30 | Q 2.6 | Page 27

Find the sum of the following geometric series:

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

Ex. 20.30 | Q 2.7 | Page 27

Find the sum of the following geometric series:

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

Ex. 20.30 | Q 2.8 | Page 27

Find the sum of the following geometric series:

x3, x5, x7, ... to n terms

Ex. 20.30 | Q 2.9 | Page 27

Find the sum of the following geometric series:

$\sqrt{7}, \sqrt{21}, 3\sqrt{7}, . . .\text { to n terms }$

Ex. 20.30 | Q 3.1 | Page 28

Evaluate the following:

$\sum^{11}_{n = 1} (2 + 3^n )$

Ex. 20.30 | Q 3.2 | Page 28

Evaluate the following:

$\sum^n_{k = 1} ( 2^k + 3^{k - 1} )$

Ex. 20.30 | Q 3.3 | Page 28

Evaluate the following:

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

Ex. 20.30 | Q 4.1 | Page 28

Find the sum of the following serie:

5 + 55 + 555 + ... to n terms;

Ex. 20.30 | Q 4.2 | Page 28

Find the sum of the following series:

7 + 77 + 777 + ... to n terms;

Ex. 20.30 | Q 4.3 | Page 28

Find the sum of the following series:

9 + 99 + 999 + ... to n terms;

Ex. 20.30 | Q 4.4 | Page 28

Find the sum of the following series:

0.5 + 0.55 + 0.555 + ... to n terms.

Ex. 20.30 | Q 4.5 | Page 28

Find the sum of the following series:

0.6 + 0.66 + 0.666 + .... to n terms

Ex. 20.30 | Q 5 | Page 28

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

Ex. 20.30 | Q 6 | Page 28

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

Ex. 20.30 | Q 7 | Page 28

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

Ex. 20.30 | Q 8 | Page 28

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

Ex. 20.30 | Q 9 | Page 28

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.

Ex. 20.30 | Q 10 | Page 28

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.

Ex. 20.30 | Q 11 | Page 28

The 4th and 7th terms of a G.P. are $\frac{1}{27} \text { and } \frac{1}{729}$ respectively. Find the sum of n terms of the G.P.

Ex. 20.30 | Q 12 | Page 28

Find the sum :

$\sum^{10}_{n = 1} \left[ \left( \frac{1}{2} \right)^{n - 1} + \left( \frac{1}{5} \right)^{n + 1} \right] .$

Ex. 20.30 | Q 13 | Page 28

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.

Ex. 20.30 | Q 14 | Page 28

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).

Ex. 20.30 | Q 15 | Page 28

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)thterm is $\frac{1}{r^n}$ .

Ex. 20.30 | Q 16 | Page 28

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) : (q − p) = 17 : 15.

Ex. 20.30 | Q 17 | Page 29

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

Ex. 20.30 | Q 18 | Page 29

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.

Ex. 20.30 | Q 19 | Page 29

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.

Ex. 20.30 | Q 20 | Page 29

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.

Ex. 20.30 | Q 21 | Page 29

Let an be the nth term of the G.P. of positive numbers.

Let $\sum^{100}_{n = 1} a_{2n} = \alpha \text { and } \sum^{100}_{n = 1} a_{2n - 1} = \beta,$ such that α ≠ β. Prove that the common ratio of the G.P. is α/β.

Ex. 20.30 | Q 22 | Page 29

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.

#### Chapter 20: Geometric Progression Exercise 20.40 solutions [Pages 39 - 40]

Ex. 20.40 | Q 1.1 | Page 39

Find the sum of the following serie to infinity:

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

Ex. 20.40 | Q 1.2 | Page 39

Find the sum of the following serie to infinity:

8 +  $4\sqrt{2}$ + 4 + ... ∞

Ex. 20.40 | Q 1.3 | Page 39

Find the sum of the following serie to infinity:

2/5 + 3/52 +2/53 + 3/54 + ... ∞.

Ex. 20.40 | Q 1.4 | Page 39

Find the sum of the following serie to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Ex. 20.40 | Q 1.5 | Page 39

Find the sum of the following serie to infinity:

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

Ex. 20.40 | Q 2 | Page 39

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

Ex. 20.40 | Q 3 | Page 40

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

Ex. 20.40 | Q 4 | Page 40

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.

Ex. 20.40 | Q 5 | Page 40

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.

Ex. 20.40 | Q 6 | Page 40

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

Ex. 20.40 | Q 7 | Page 40

Find the rational number whose decimal expansion is $0 . 423$.

Ex. 20.40 | Q 8.1 | Page 40

Find the rational numbers having the following decimal expansion:

$0 . \overline3$

Ex. 20.40 | Q 8.2 | Page 40

Find the rational numbers having the following decimal expansion:

$0 .\overline {231 }$

Ex. 20.40 | Q 8.3 | Page 40

Find the rational numbers having the following decimal expansion:

$3 . 5\overline 2$

Ex. 20.40 | Q 8.4 | Page 40

Find the rational numbers having the following decimal expansion:

$0 . 6\overline8$

Ex. 20.40 | Q 9 | Page 40

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.

Ex. 20.40 | Q 10 | Page 40

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

Ex. 20.40 | Q 11 | Page 40

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.

Ex. 20.40 | Q 12 | Page 40

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.

Ex. 20.40 | Q 13 | Page 40

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

$\frac{2S S_1}{S^2 + S_1}\text { and } \frac{S^2 - S_1}{S^2 + S_1}$

#### Chapter 20: Geometric Progression Exercise 20.50 solutions [Pages 45 - 46]

Ex. 20.50 | Q 1 | Page 45

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

Ex. 20.50 | Q 2 | Page 45

If a, b, c are in G.P., prove that $\frac{1}{\log_a m}, \frac{1}{\log_b m}, \frac{1}{\log_c m}$ are in A.P.

Ex. 20.50 | Q 3 | Page 45

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

Ex. 20.50 | Q 4 | Page 45

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.

Ex. 20.50 | Q 5 | Page 45

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.

Ex. 20.50 | Q 6 | Page 45

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.

Ex. 20.50 | Q 7 | Page 46

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.

Ex. 20.50 | Q 8.1 | Page 46

If a, b, c are in G.P., prove that:

a (b2 + c2) = c (a2 + b2)

Ex. 20.50 | Q 8.2 | Page 46

If a, b, c are in G.P., prove that:

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

Ex. 20.50 | Q 8.3 | Page 46

If a, b, c are in G.P., prove that:

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

Ex. 20.50 | Q 8.4 | Page 46

If a, b, c are in G.P., prove that:

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

Ex. 20.50 | Q 8.5 | Page 46

If a, b, c are in G.P., prove that:

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

Ex. 20.50 | Q 9.1 | Page 46

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

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

Ex. 20.50 | Q 9.2 | Page 46

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

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

Ex. 20.50 | Q 9.3 | Page 46

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

(b + c) (b + d) = (c + a) (c + d)

Ex. 20.50 | Q 10.1 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a2, b2, c2

Ex. 20.50 | Q 10.2 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a3, b3, c3

Ex. 20.50 | Q 10.3 | Page 46

If a, b, c are in G.P., prove that the following is also in G.P.:

a2 + b2, ab + bc, b2 + c2

Ex. 20.50 | Q 11.1 | Page 46

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

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

Ex. 20.50 | Q 11.2 | Page 46

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

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

Ex. 20.50 | Q 11.3 | Page 46

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

$\frac{1}{a^2 + b^2}, \frac{1}{b^2 - c^2}, \frac{1}{c^2 + d^2} \text { are in G . P } .$

Ex. 20.50 | Q 11.4 | Page 46

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

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

Ex. 20.50 | Q 12 | Page 46

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

Ex. 20.50 | Q 13 | Page 46

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

$\frac{a^2 + ab + b^2}{bc + ca + ab} = \frac{b + a}{c + b}$
Ex. 20.50 | Q 14 | Page 46

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.

Ex. 20.50 | Q 15 | Page 46

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

Ex. 20.50 | Q 16 | Page 46

If pth, qth, rth and sth terms of an A.P. be in G.P., then prove that p − q, q − r, r − s are in G.P.

Ex. 20.50 | Q 17 | Page 46

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.

Ex. 20.50 | Q 18 | Page 46

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

Ex. 20.50 | Q 19 | Page 46

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.

Ex. 20.50 | Q 20 | Page 46

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.

Ex. 20.50 | Q 21 | Page 46

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

Ex. 20.50 | Q 22 | Page 46

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.

Ex. 20.50 | Q 23 | Page 46

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

#### Chapter 20: Geometric Progression Exercise 20.60 solutions [Pages 54 - 55]

Ex. 20.60 | Q 1 | Page 54

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

Ex. 20.60 | Q 2 | Page 54

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

Ex. 20.60 | Q 3 | Page 54

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

Ex. 20.60 | Q 4.1 | Page 55

Find the geometric means of the following pairs of number:

2 and 8

Ex. 20.60 | Q 4.2 | Page 55

Find the geometric means of the following pairs of number:

a3b and ab3

Ex. 20.60 | Q 4.3 | Page 55

Find the geometric means of the following pairs of number:

−8 and −2

Ex. 20.60 | Q 5 | Page 55

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

Ex. 20.60 | Q 6 | Page 55

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

Ex. 20.60 | Q 7 | Page 55

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

Ex. 20.60 | Q 8 | Page 55

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio $(3 + 2\sqrt{2}) : (3 - 2\sqrt{2})$ .

Ex. 20.60 | Q 9 | Page 55

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

Ex. 20.60 | Q 10 | Page 55

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

Ex. 20.60 | Q 11 | Page 55

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.

Ex. 20.60 | Q 12 | Page 55

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

$a : b = (2 + \sqrt{3}) : (2 - \sqrt{3}) .$

Ex. 20.60 | Q 13 | Page 55

If one A.M., A and two geometric means G1 and G2 inserted between any two positive numbers, show that $\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A$

#### Chapter 20: Geometric Progression solutions [Page 56]

Q 1 | Page 56

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

Q 2 | Page 56

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

Q 3 | Page 56

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

Q 4 | Page 56

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.

Q 5 | Page 56

If pth, qth and rth terms of a G.P. re x, y, z respectively, then write the value of xq − r yr − pzp − q.

Q 6 | Page 56

If A1, A2 be two AM's and G1G2 be two GM's between and b, then find the value of $\frac{A_1 + A_2}{G_1 G_2}$

Q 7 | Page 56

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.

Q 8 | Page 56

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

Q 9 | Page 56

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

Q 10 | Page 56

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

#### Chapter 20: Geometric Progression solutions [Pages 57 - 58]

Q 1 | Page 57

If in an infinite G.P., first term is equal to 10 times the sum of all successive terms, then its common ratio is

• 1/10

• 1/11

• 1/9.

• 1/20

Q 2 | Page 57

If the first term of a G.P. a1a2a3, ... is unity such that 4 a2 + 5 a3 is least, then the common ratio of G.P. is

• −2/5

• −3/5

• 2/5

•  none of these

Q 3 | Page 57

If abc are in A.P. and xyz are in G.P., then the value of xb − c yc − a za − b is

•  0

• 1

•  xyz

•  xa yb zc

Q 4 | Page 57

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

•  4

•  8

Q 5 | Page 57

If abc are in G.P. and a1/b1/y = c1/z, then xyz are in

• (a) AP

• (b) GP

• (c) HP

• (d) none of these

Q 6 | Page 57

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

Q 7 | Page 57

The fractional value of 2.357 is

• (a) 2355/1001

• (b) 2379/997

• (c) 2355/999

• (d) none of these

Q 8 | Page 57

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

Q 9 | Page 57

The value of 91/3 . 91/9 . 91/27 ... upto inf, is

• (a) 1

• (b) 3

• (c) 9

• (d) none of these

Q 10 | Page 57

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

Q 11 | Page 57

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

Q 12 | Page 57

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

Q 13 | Page 57

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

Q 14 | Page 57

If abc are in G.P. and xy are AM's between ab 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) $\frac{1}{x} + \frac{1}{y} = \frac{2}{b}$

Q 15 | Page 58

If A be one A.M. and pq 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) $\frac{pq}{2}$

Q 16 | Page 58

If pq be two A.M.'s and G be one G.M. between two numbers, then G2

• (a) (2p − q) (p −  2q)

• (b) (2p − q) (2q − p)

• (c) (2p − q) (p + 2q)

• (d) none of these

Q 17 | Page 58

If x is positive, the sum to infinity of the series $\frac{1}{1 + x} - \frac{1 - x}{(1 + x )^2} + \frac{(1 - x )^2}{(1 + x )^3} - \frac{(1 - x )^3}{(1 + x )^4} + . . . . . . is$

• (a) 1/2

• (b) 3/4

• (c) 1

• (d) none of these

Q 18 | Page 58

If x = (43) (46) (46) (49) .... (43x) = (0.0625)−54, the value of x is

• (a) 7

• (b) 8

• (c) 9

• (d) 10

Q 19 | Page 58

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

• (a)

• (b) x + 1

• (c) $\frac{x}{2x + 1}$

• (d) $\frac{x + 1}{2x + 1}$

Q 20 | Page 58

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) $\frac{1}{5}$

• (b) $\frac{1}{5}$

• (c) 4

• (d) none of these

Q 21 | Page 58

Let x be the A.M. and yz 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

Q 22 | Page 58

The product (32), (32)1/6 (32)1/36 ... to ∞ is equal to

• (a) 64

• (b) 16

• (c) 32

• (d) 0

Q 23 | Page 58

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

Q 24 | Page 58

In a G.P. if the (m + n)th term is p and (m − n)th term is q, then its mth term is

• (a) 0

• (b) pq

• (c) $\sqrt{pq}$

• (d) $\frac{1}{2}(p + q)$

Q 25 | Page 58

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

## Chapter 20: Geometric Progression

Ex. 20.10Ex. 20.20Ex. 20.30Ex. 20.40Ex. 20.50Ex. 20.60Others

## RD Sharma solutions for Class 11 Mathematics chapter 20 - Geometric Progression

RD Sharma solutions for Class 11 Maths chapter 20 (Geometric Progression) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 11 Mathematics chapter 20 Geometric Progression are Sum to N Terms of Special Series, Introduction of Sequence and Series, Concept of Sequences, Concept of Series, Arithmetic Progression (A.P.), Geometric Progression (G. P.), Relationship Between A.M. and G.M..

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