MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 14

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

Important Questions on Geometric Progressions

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 15

Show that the ratio of the sum of first n terms of a

G . P .

to the sum of terms from

{(n + 1)}^{t h}

to

{(2 n)}^{t h}

term is

\frac{1}{r^{n}}

.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 16

If

a

and

b

are the roots of

x^{2} - 3 x + p = 0

and

c, d

are the roots

x^{2} - 12 x + q = 0

, where

a, b, c, d

form a

G . P .

Prove that

(q + p) : (q - p) = 17 : 15

.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 17

How many terms of the G.P.

3, \frac{3}{2}, \frac{3}{4} \dots \dots

are needed to give the sum

\frac{3069}{512}

?

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 18

A person has

2

parents,

4

grandparents,

8

great grand parents, and so on. Find the number of his ancestors during the ten generations preceding his own.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 19

If

S_{1}, S_{2}, \dots ., 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} + 2 S_{3} + 3 S_{4} + . . . + (n - 1) S_{n} = l^{n} + 2^{n} + 3^{n} + . . . + n^{n}

.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 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 .

.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 21

Let

a_{n}

be the

n t h

term of the

G . P .

of positive numbers. Let

\sum_{n = 1}^{100} a_{2 n} = α

and

\sum_{n = 1}^{100} a_{2 n - 1} = β

, such that

a \neq β

. Prove that the common ratio of the

G . P .

is

\frac{α}{β}

.

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MATHEMATICS for CLASS XI>Chapter 20 - Geometric Progressions>EXERCISE 20.3>Q 22

Find the sum of

2 n

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.