Question

Given a G.P. with a = 729 and 7 th term 64, determine S 7 .

Answer 1

The first term of the G.P. is 729 and the 7 th term of the G.P. is 64.

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

Here,

a=729

The formula for n th term of a G.P. is given by,

a n =a r n−1 (1)

Substitute the value of a to obtain the 7 th term.

a 7 =729 r 7−1 64=729 r 6 r 6 = 64 729 r 6 = ( 2 3 ) 6

Compare the terms on both the sides.

r= 2 3

Here, r<1

The formula for the sum of first n terms of a G.P. for r<1 is given by,

S n = a( 1− r n ) 1−r

Substitute the values of a and r in equation (1) to obtain the 7 th term.

S 7 = 729( 1− ( 2 3 ) 7 ) 1−( 2 3 ) S 7 = ( 3 ) 6 ( 3 7 − 2 7 3 7 ) 1 3 = ( 3 ) 7 ( ( 3 ) 7 − ( 2 ) 7 3 7 ) = ( 3 ) 7 − ( 2 ) 7

Further simplify the above expression.

S 7 =2187−128 =2059

Thus, sum of the7 terms of the G.P. is ( 3 ) 7 − ( 2 ) 7 .