## GEOMETRIC SERIES

Geometric series :

A series whose terms are in geometric progression is called geometric series.

To find the sum of n terms of the geometric series, we use one of the formulas given below.

sn  =  a(rn - 1)/(r - 1)    if r > 1

sn  =  a(1 - rn)/(1 - r)  if r < 1

sn  =  a/(1 - r)   if r  =  1

Example 1 :

Find the sum of the series

1 + 3 + 9 + ................. to 10 terms

Solution :

a  =  1, r  =  3 > 1 and n  =  10

sn  =  a(rn - 1)/(r - 1)    if r > 1

s10  =  1(310 - 1)/(3 - 1)

s10  =  (59049 - 1)/2

s10  =  59048/2

s10  =  29524

Example 2 :

A ball is dropped from a height of 6 m and on each bounce it bounces 2/3 of its previous height.

(i) What is the total length of the downward paths ?

(ii) What is the total length of the upward paths ?

(iii) How far does the ball travel till it stops bouncing?

Solution :

Distance covered in the downward path

=  6 + 4 + 8/3 + 16/9 + .............

a  =  6 and r  =  4/6 ==>  2/3

To find the sum of infinite series, we use the formula

sn  =  a / (1 - r)

sn  =  6 / (1 - (2/3))

sn  =  6 / (1/3)

sn  =  18 m

(ii)  Distance covered in the upward path

=  4 + 8/3 + 16/9 + ............

Here a  =  4 and r  =  2/3

sn  =  a / (1 - r)

sn  =  4 / (1 - (2/3))

sn  =  4 / (1/3)

sn  =  12 m

(iii) Total distance covered  =  18 + 12

=  30 m After having gone through the stuff given above, we hope that the students would have understood how to find the sum of geometric series.

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