#### Chapters

Chapter 2: Relations

Chapter 3: Functions

Chapter 4: Measurement of Angles

Chapter 5: Trigonometric Functions

Chapter 6: Graphs of Trigonometric Functions

Chapter 7: Values of Trigonometric function at sum or difference of angles

Chapter 8: Transformation formulae

Chapter 9: Values of Trigonometric function at multiples and submultiples of an angle

Chapter 10: Sine and cosine formulae and their applications

Chapter 11: Trigonometric equations

Chapter 12: Mathematical Induction

Chapter 13: Complex Numbers

Chapter 14: Quadratic Equations

Chapter 15: Linear Inequations

Chapter 16: Permutations

Chapter 17: Combinations

Chapter 18: Binomial Theorem

Chapter 19: Arithmetic Progression

Chapter 20: Geometric Progression

Chapter 21: Some special series

Chapter 22: Brief review of cartesian system of rectangular co-ordinates

Chapter 23: The straight lines

Chapter 24: The circle

Chapter 25: Parabola

Chapter 26: Ellipse

Chapter 27: Hyperbola

Chapter 28: Introduction to three dimensional coordinate geometry

Chapter 29: Limits

Chapter 30: Derivatives

Chapter 31: Mathematical reasoning

Chapter 32: Statistics

Chapter 33: Probability

#### RD Sharma Mathematics Class 11

## Chapter 20 : Geometric Progression

#### Pages 9 - 10

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

4, −2, 1, −1/2, ...

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

−2/3, −6, −54, ...

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

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

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, ...

Show that the sequence <*a*_{n}>, defined by a_{n} = \[\frac{2}{3^n}\], n ϵ N is a G.P.

Find:

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

Find:

the 10^{th} term of the G.P.

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

Find :

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

Find :

the 12th term of the G.P.

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

Find :

nth term of the G.P.

\[\sqrt{3}, \frac{1}{\sqrt{3}}, \frac{1}{3\sqrt{3}}, . . .\]

Find :

the 10^{th} term of the G.P.

\[\sqrt{2}, \frac{1}{\sqrt{2}}, \frac{1}{2\sqrt{2}}, . . .\]

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

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

Which term of the G.P. :

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

Which term of the G.P. :

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

Which term of the G.P. :

\[\sqrt{3}, 3, 3\sqrt{3}, . . . \text { is } 729 ?\]

Which term of the G.P. :

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

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

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

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

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

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

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

If 5^{th}, 8^{th} and 11^{th} terms of a G.P. are p. q and s respectively, prove that q^{2} = ps.

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

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

If a, b, c, d and p are different real numbers such that:

(a^{2} + b^{2} + c^{2}) p^{2} − 2 (ab + bc + cd) p + (b^{2} + c^{2} + d^{2}) ≤ 0, then show that a, b, c and d are in G.P.

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.

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

#### Page 16

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

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

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

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.

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.

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.

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.

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

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

#### Pages 27 - 29

Find the sum of the following geometric progression:

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

Find the sum of the following geometric progression:

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

Find the sum of the following geometric progression:

1, −1/2, 1/4, −1/8, ... to 9 terms;

Find the sum of the following geometric progression:

(a^{2} − b^{2}), (a − b), \[\left( \frac{a - b}{a + b} \right)\] to n terms;

Find the sum of the following geometric progression:

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

Find the sum of the following geometric series:

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

Find the sum of the following geometric series:

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

Find the sum of the following geometric series:

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

Find the sum of the following geometric series:

(x +y) + (x^{2} + xy + y^{2}) + (x^{3} + x^{2}y + xy^{2} + y^{3}) + ... to n terms;

Find the sum of the following geometric series:

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

Find the sum of the following geometric series:

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

Find the sum of the following geometric series:

1, −a, a^{2}, −a^{3}, ... to n terms (a ≠ 1)

Find the sum of the following geometric series:

x^{3}, x^{5}, x^{7}, ... to n terms

Find the sum of the following geometric series:

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

Evaluate the following:

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

Evaluate the following:

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

Evaluate the following:

\[\sum^{10}_{n = 2} 4^n\]

Find the sum of the following serie:

5 + 55 + 555 + ... to *n* terms;

Find the sum of the following series:

7 + 77 + 777 + ... to n terms;

Find the sum of the following series:

9 + 99 + 999 + ... to n terms;

Find the sum of the following series:

0.5 + 0.55 + 0.555 + ... to n terms.

Find the sum of the following series:

0.6 + 0.66 + 0.666 + .... to n terms

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

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

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

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

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.

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.

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.

Find the sum :

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

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.

If S_{1}, S_{2}, S_{3} be respectively the sums of n, 2n, 3n terms of a G.P., then prove that \[S_1^2 + S_2^2\] = *S*_{1} (*S*_{2} + *S*_{3}).

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 \[\frac{1}{r^n}\] .

If a and b are the roots of x^{2} − 3x + p = 0 and c, d are the roots x^{2} − 12x + q = 0, where a, b, c, d form a G.P. Prove that (q + p) : (q − p) = 17 : 15.

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

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.

If S_{1}, S_{2}, ..., S_{n} 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 S_{1} + S_{2} + 2S_{3} + 3S_{4} + ... (n − 1) S_{n} = 1^{n}^{ }+ 2^{n} + 3^{n} + ... + n^{n}.

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 a_{n} 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.

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.

#### Pages 39 - 40

Find the sum of the following serie to infinity:

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

Find the sum of the following serie to infinity:

8 + \[4\sqrt{2}\] + 4 + ... ∞

Find the sum of the following serie to infinity:

2/5 + 3/5^{2} +2/5^{3} + 3/5^{4} + ... ∞.

Find the sum of the following serie to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Find the sum of the following serie to infinity:

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

Prove that: (9^{1/3} . 9^{1/9} . 9^{1/27} ... ∞) = 3.

Prove that: (2^{1/4} . 4^{1/8} . 8^{1/16}. 16^{1/32} ... ∞) = 2.

If *S*_{p} denotes the sum of the series 1 + r^{p} + r^{2p} + ... to ∞ and s_{p} the sum of the series 1 − r^{p} + r^{2p} − ... to ∞, prove that S_{p} + s_{p} = 2 . S_{2p}.

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.

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

Find the rational number whose decimal expansion is \[0 . 423\].

Find the rational numbers having the following decimal expansion:

\[0 . \overline3\]

Find the rational numbers having the following decimal expansion:

\[0 .\overline {231 }\]

Find the rational numbers having the following decimal expansion:

\[3 . 5\overline 2\]

Find the rational numbers having the following decimal expansion:

\[0 . 6\overline8\]

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.

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

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.

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.

If *S* denotes the sum of an infinite G.P. *S*_{1} denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively

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

#### Pages 45 - 46

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

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.

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

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.

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.

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.

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.

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

a (b^{2} + c^{2}) = c (a^{2} + b^{2})

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

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

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

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

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

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

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

(a + 2b + 2c) (a − 2b + 2c) = a^{2} + 4c^{2}.

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

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

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}

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

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

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

a^{2}, b^{2}, c^{2}

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

a^{3}, b^{3}, c^{3}

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

a^{2} + b^{2}, ab + bc, b^{2} + c^{2}

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

(a^{2} + b^{2}), (b^{2} + c^{2}), (c^{2} + d^{2}) are in G.P.

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

(a^{2} − b^{2}), (b^{2} − c^{2}), (c^{2} − d^{2}) are in G.P.

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

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

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

(a^{2} + b^{2} + c^{2}), (ab + bc + cd), (b^{2} + c^{2} + d^{2}) are in G.P.

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

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

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

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.

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.

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.

If x^{a} = x^{b}^{/2} z^{b}^{/2} = z^{c}, then prove that \[\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\] are in A.P.

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.

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x^{2}, b^{2}, y^{2} are in A.P.

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.

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.

If p^{th}, q^{th} and r^{th }terms of an A.P. and G.P. are both a, b and c respectively, show that \[a^{b - c} b^{c - a} c^{a - b} = 1\]

#### Pages 54 - 55

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

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

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

Find the geometric means of the following pairs of number:

2 and 8

Find the geometric means of the following pairs of number:

a^{3}b and ab^{3}

Find the geometric means of the following pairs of number:

−8 and −2

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

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

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

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

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

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

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.

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

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

If one A.M., A and two geometric means G_{1} and G_{2} inserted between any two positive numbers, show that \[\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A\]

#### Page 56

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

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

If log_{x}a, a^{x}^{/2} and log_{b} x are in G.P., then write the value of x.

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.

*If p ^{th}, q^{th} and r^{th} terms of a G.P. re x, y, z respectively, then write the value of x^{q}^{ − r} y^{r}^{ − p}z^{p}^{ − q}.*

If *A*_{1}, A_{2} be two AM's and *G*_{1}, *G*_{2} be two GM's between *a *and *b*, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]

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.

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

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

If *a* = 1 + *b* + *b*^{2} + *b*^{3} + ... to ∞, then write *b* in terms of *a*.

#### Pages 57 - 58

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

If the first term of a G.P. *a*_{1}, *a*_{2}, *a*_{3}, ... is unity such that 4 *a*_{2} + 5 *a*_{3} is least, then the common ratio of G.P. is

(a) −2/5

(b) −3/5

(c) 2/5

(d) none of these

If *a*, *b*, *c* are in A.P. and *x*, *y*, *z* are in G.P., then the value of *x ^{b}*

^{ − c}

*y*

^{c}^{ − a}

*z*

^{a}^{ − b}is

(a) 0

(b) 1

(c) *xyz*

(d) *x ^{a}*

*y*

^{b}*z*

^{c}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

If *a*, *b*, *c* are in G.P. and *a*^{1}^{/x }= *b*^{1}^{/y} = *c*^{1}^{/z}, then *xyz* are in

(a) AP

(b) GP

(c) HP

(d) none of these

If *S* be the sum, *P* the product and *R* be the sum of the reciprocals of *n* terms of a GP, then *P*^{2} is equal to

(a) *S*/*R*

(b) *R*/*S*

(c) (*R*/*S*)^{n}

(d) (*S*/*R*)^{n}

The fractional value of 2.357 is

(a) 2355/1001

(b) 2379/997

(c) 2355/999

(d) none of these

If *p*th, *q*th and *r*th 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

The value of 9^{1/3} . 9^{1/9} . 9^{1/27} ... upto inf, is

(a) 1

(b) 3

(c) 9

(d) none of these

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

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

The *n*th 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

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

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) \[\frac{1}{x} + \frac{1}{y} = \frac{2}{b}\]

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) \[\frac{pq}{2}\]

If *p*, *q* be two A.M.'s and *G* be one G.M. between two numbers, then *G*^{2} =

(a) (2*p* −* q*) (*p* − 2*q*)

(b) (2*p* − *q*) (2*q* − *p*)

(c) (2*p* − *q*) (*p* + 2*q*)

(d) none of these

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

If *x* = (4^{3}) (4^{6}) (4^{6}) (4^{9}) .... (4^{3x}) = (0.0625)^{−54}, the value of *x* is

(a) 7

(b) 8

(c) 9

(d) 10

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

(a) *x *

(b) *x* + 1

(c) \[\frac{x}{2x + 1}\]

(d) \[\frac{x + 1}{2x + 1}\]

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

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

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

(a) 64

(b) 16

(c) 32

(d) 0

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

In a G.P. if the (*m* +* n*)* ^{th}* term is

*p*and (

*m*−

*n*)

*term is*

^{th}*q*, then its

*m*

^{th}term is

(a) 0

(b) *pq*

(c) \[\sqrt{pq}\]

(d) \[\frac{1}{2}(p + q)\]

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 *p*^{2}*R*^{3} : *S*^{3} is equal to

(a) 1 : 1

(b) (Common ratio)^{n} : 1

(c) (First term)^{2} : (Common ratio)^{2}

(d) None of these

#### RD Sharma Mathematics Class 11

#### Textbook solutions for Class 11

## 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 Introduction of Sequence and Series, Sum to N Terms of Special Series, Relationship Between A.M. and G.M., Geometric Progression (G. P.), Arithmetic Progression (A.P.), Concept of Series, Concept of Sequences.

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