RD Sharma solutions for Mathematics [English] Class 11 chapter 20 - Geometric Progression [Latest edition]

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 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.: `sqrt3, 3, 3sqrt3`, ... 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.

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

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.

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

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 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 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:

(a² − b²), (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) + (x² + xy + y²) + (x³ + x²y + xy² + y³) + ... to n terms;

Find the sum of the following geometric series:

`3/5 + 4/5^2 + 3/5^3 + 4/5^4 + ....` 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, a², −a³, ....to n terms (a ≠ 1)

Find the sum of the following geometric series:

x³, x⁵, x⁷, ... to n terms

Find the sum of the following geometric series:

`sqrt7, sqrt21, 3sqrt7,...` 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 sequence \[\sqrt{3}, 3, 3\sqrt{3},\]  ... must be taken to make the sum \[39 + 13\sqrt{3}\] ?

The ratio of the sum of the first three terms to that of the 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 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).

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 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.

How many terms of the G.P. `3, 3/2, 3/4` ..... are needed to give the sum `3069/512`?

If S₁, S₂, ..., 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 S₁ + S₂ + 2S₃ + 3S₄ + ... (n − 1) S_n = 1ⁿ + 2ⁿ + 3ⁿ + ... + nⁿ.

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.

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 series to infinity:

10 − 9 + 8.1 − 7.29 + ... ∞

Find the sum of the following series to infinity:

`1/3+1/5^2 +1/3^3+1/5^4 + 1/3^5 + 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.

Find the rational number whose decimal expansion is `0.4bar23`.

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. S₁ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively

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

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.

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 (b² + c²) = c (a² + b²)

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) = a² + 4c².

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.:

a², b², c²

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

a³, b³, c³

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

a² + b², ab + bc, b² + c²

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

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

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

(a² − b²), (b² − c²), (c² − d²) 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:

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

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

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:

a³b and ab³

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.

The sum of two numbers is 6 times their geometric means, show that the numbers are in the ratio `(3+2sqrt2):(3-2sqrt2)`.

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

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 G₁ and G₂ inserted between any two positive numbers, show that \[\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A\]

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 − pz^p − q.

If A₁, A₂ be two AM's and G₁, G₂ 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 Aand G respectively. 

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

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

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

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

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

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 

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

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

The fractional value of 2.357 is 

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

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

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 

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 

The n^th term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then its first term is ______.

If second term of a G.P. is 2 and the sum of its infinite terms is 8, then its first term is

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

If A be one A.M. and p, q be two G.M.'s between two numbers, then 2 A is equal to 

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

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

If x = (4³) (4⁶) (4⁶) (4⁹) .... (4^3x) = (0.0625)⁻⁵⁴, the value of x is 

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

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 

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 

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

The two geometric means between the numbers 1 and 64 are 

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

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