KC Sinha Mathematics Solution Class 11 Chapter 18 Geometric Progressions Exercise 18.2

 Exercise 18.2

page no 18.14

Type 1

Question 1

Find the sum of indicated terms of each of the following Geometric progressions:

(i) $1,2,4,8 \ldots, 12$ terms

(ii) $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots, n$ terms

(iii) $0.15,0.015,0.0015, \ldots ., 20$ terms

(iv) $1,-\frac{1}{2}, \frac{1}{4},-\frac{1}{8}, \ldots, n$ terms

(v) $1,3,9,27, \ldots ; 8$ terms

(vi) $1,-3,9,-27, \ldots ; 9$ terms

(vii) $x^{2}, x^{4}, x^{6}, \ldots ; n$ terms $(x \neq \pm 1)$

(viii) $1,-a, a^{2},-a^{3}, \ldots, n$ terms $(a \neq-1)$

(ix) $1+\frac{2}{3}+\frac{4}{9}+\ldots ; n$ terms and 5 terms

Question 2

A G.P. has first term 729 and 7 th term 64. Find the sum of its first 7 terms

Question 3

Find the sum of the products of the corresponding terms of the sequence $2,4,8,16,32$ and $128,32,8,2, \frac{1}{2}$,

Question 4

Find the sum to $n$ terms of the series

$\left(x^{2}+\frac{1}{x^{2}}+2\right)+\left(x^{4}+\frac{1}{x^{4}}+5\right)+\left(x^{6}+\frac{1}{x^{6}}+8\right)+\ldots$

Question 5

Evaluate $\sum_{k=1}^{n}\left(2^{k}+3^{k}-1\right)$.

Question 6

How many terms of the series $1+2+2^{2}+\ldots$ must be taken to make 511 ?

Question 7

How many terms of the series $3+\frac{3}{2}+\frac{3}{4}+\ldots$ must be taken to make $\frac{3063}{512}$

Question 8

The sum of some terms of a G.P. is 315 whose first term and the common ratio are 5 and 2 respectively. Find the last term and the number of terms

Question 9

A G.P. consists of an even number of terms. If the sum of all the tems is 5 times the sum of terms occupying odd places, then find its common twio

Question 10

Find the sum of the series :

(i) $6+66+666+\ldots$ to $n$ terms

(ii) $9+99+999+\ldots$ to $n$ terms

(iii) $4+44+444+\ldots$ to $n$ terms

(iv) $5+55+555+\ldots$ to $n$ terms

(v) $7+77+777+\ldots$ to $n$ terms

(vi) $0.6+0.66+0.666+\ldots$ to $n$ terms

(vii) $0.5+0.55+0.555+\ldots$ to $n$ terms

page no 18.15

Question 11

Show that the ratio of the sum of first $n$ terms of a G.P. to the sum of next $n$ terms is $\frac{1}{r^{n}}$, where $r$ is the common ratio of G.P.

Question 12

Let $S$ be the sum, $P$ the product and $R$ the sum of reciprocals of $n$ terms of a G.P. Prove that $P^{2} R^{n}=S^{n}$.

$[H O T S]$