# Sum of Geometric progression(G.P)

**How to find sum of n term of any Geometric progression**

If first term is a and common ratio is r then series will be

S_{n}=a+ar+ar^{2}+ar^{3}+……….+ar^{n-1} ------------(1)

Multiply r on both side then we get

r*S_{n}=ar+ar^{2}+ar^{3}+ar^{4}……….+ar^{n-1}+ar^{n} --------(2)

Subtracting (2) from (1)

S_{n}- r*S_{n} =a+ar+ar^{2}+ar^{3}+……….+ar^{n-1}- ar-ar^{2}-ar^{3}-ar^{4}……….- ar^{n-1}-ar^{n}

s_{n}(1-r)=a-ar^{n}=a(1-r^{n})

s_{n}= a(1-r^{n})/(1-r)

s_{n}=a(r^{n}-1)/(r-1)

based on this result we can find sum of any Geometric series

for example

1+2+4+8+16………to 20 term

We have to find sum of 20 term

Here a=1

r=2 and number of term n=20

we know s_{n}= a(r^{n}-1)/(r-1)

s_{20}=1(2^{20}-1)/(2-1)

s_{20}= 2^{20}-1

## Properties of Geometric progression(G.P)

(1)Each term multiply or divide by non Zero then obtained series is also in G.P

(2)Reciprocal of each term of geometric series is also in geometric series.

For example

If series 2,4,8,16……… is in gp then reciprocal of all term

1/2,1/4,1/8,1/16 is also in gp