# EXERCISE 20.3

QUESTION 1

Find the sum of the following geometric progressions:

(i) to 7 terms;

Sol :

Here, a=2 and r=3

= 2787 – 1

= 2186

(ii) to 8 terms

Sol :

Here, a=1 and r=3

= 3280

(iii) to 9 terms;

Sol :

Here, a=1 and

(iv) to n terms

Sol :

Here, and

(v) to 10 terms.

Sol :

Here, a=4 and

QUESTION 2

Find the sum of the following geometric series:

(i) to 8 terms;

Sol :

Here, a=0.15 and

(ii) to 8 terms

Sol :

Here, and

(iii) to 5 terms;

Sol :

Here, and

(iv) to n terms

Sol :

(v)  to 2 terms

Sol :

(vi)

Sol :

(vii) to n terms

Sol :

(viii) to n terms

Sol :

(ix) to n terms

Sol :

QUESTION 3

Evaluate the following :

(i)

Sol :

= 22 + 265719

= 265741

(ii)

Sol :

(iii)

Sol :

QUESTION 4

Find the sum of the following series:

(i) to terms;

Sol :

taking 5 as common :

terms

terms

times

(ii) to terms

Sol :

Taking 7 as common :

terms

n terms

times

(iii) to terms

Sol :

This can be rewritten as :

times

(iv) to terms.

Sol :

Taking 5 as common :

terms

to n terms )

=\dfrac{5}{9}\left\{n-\left(\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dots{\text{ n terms }}\right)\right}

(v) to terms

Sol :

Taking 6 as common :

=\dfrac{6}{9}\left\{n-\left(\dfrac{1}{10}+\dfrac{1}{10^2}+\dfrac{1}{10^3}+\dots{\text{ n terms }}\right)\right}

QUESTION 5

How many terms of the G.P. be taken together to make ?

Sol :

First term , a = 3

Common ratio,

n = 10

QUESTION 6

How many terms of the series must be taken to make the sum equal to 728 ?

Sol :

First term , a = 2

Common ratio, r = 3

Given :

n = 6

QUESTION 7

How many terms of the sequence must be taken to make the sum ?

Sol :

First term ,

Common ratio,

n = 6

QUESTION 8

The sum of n terms of the G.P. is 381 . Find the value of n

Sol :

First term , a = 3

Common ratio, r = 3

Sum of n terms,

n = 7

QUESTION 9

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.

Sol :

common ratio, r = 3

term,

(i)

Now,

[from(i)]

a = 1458 – 1456

a = 2

QUESTION 10

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.

Sol :

Let a be the first term and r be the common ratio of the G.P.

and

Then, according to the question

Now, let

or

or

or

But, is not possible.

QUESTION 11

The 4 th and 7 th terms of a G.P. are and respectively. Find the sum of n terms of the G.P.

Sol :

Let a be the first term and r be the common ratio of the G.P.

QUESTION 12

QUESTION 13

QUESTION 14

QUESTION 15

QUESTION 16