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+ar²+ar³+……….+ar^n-1 ------------(1)

Multiply r on both side then we get
r*S_n=ar+ar²+ar³+ar⁴……….+ar^n-1+arⁿ --------(2)

Subtracting (2) from (1)

S_n- r*S_n =a+ar+ar²+ar³+……….+ar^n-1- ar-ar²-ar³-ar⁴……….- ar^n-1-arⁿ

s_n(1-r)=a-arⁿ=a(1-rⁿ)

s_n= a(1-rⁿ)/(1-r)

s_n=a(rⁿ-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(rⁿ-1)/(r-1)

s₂₀=1(2²⁰-1)/(2-1)

s₂₀= 2²⁰-1

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