## Geometric Series

In this page geometric series we are going to see the formula to find sum of the geometric series and example problems with detailed steps.We have three formulas to find the sum of the series.One of the formula will be used depending upon the value of the common ratio (r) that we get.

• sn = a( 1- rn)⁄( 1 - r)
if r < 1

• sn = a( rn - 1 )⁄( r - 1)
if r > 1

• sn = na
if r = 1

• Sum of infinite series sn = a ⁄( 1 - r)

Example 1:

Find the sum of the series 1 + 3 + 9 + ................. to 10 terms

Solution:

a = 1,r = 3 and n = 10

Here the common ratio r is greater than 1

so we need to use the first formula  geometric series

sn = a( r - 1n)⁄( r - 1)

sn = 1( 310 - 1)⁄( 3 - 1)

s10 = ( 310 - 1)⁄ 2
Therefore the sum of 10 terms in the G.P is 29524

Example 2:

A ball is dropped from a heigh 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 + .............

This is a geometric-series. The first term is 6 and the common  ratio = 2/3

Here we don't know the last term of the sequence.So we can consider this as infinite series

sn = a ⁄( 1 - r)

= 6 / (1-2/3)

= 6 / (3-2/3)

= 6 x (3/1)

= 18 m

(ii)  Distance covered in the upward path = 4 + 8/3 + 16/9 + ............

Here a = 4 and r = 2/3

again we need to use the same formula

sn = a ⁄( 1 - r)

= 4 / (1-2/3)

= 4 / (3-2/3)

= 4 x (3/1)

= 12 m

(iii) Total distance covered = 18 + 12

= 30 m

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: