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:

It subtracts sadness and adds happiness in our life.

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”

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