# Geometric progression(G.P)

**What is Geometric progression**

Series of number is known as geometric series or geometric progression if and only if ratio of any term and next term are same or constant for all term.

same or constant ratio is known as common ratio of G.P

for example 1,2,2^{2},2^{3},2^{4}…….(common ratio=2)

3,9,27,81………. (common ratio=3)

## How to find nth term or General term of G.P

If a is first term and r is common ratio then
term of Geometric progression will be

t_{1}=a

t_{2}=ar=ar^{2-1}

t_{3}=ar2=ar^{3-1}

t_{4}=ar3=ar^{4-1}

t_{5}=ar4=ar^{5-1}

and so on.

t_{n}=ar^{n-1}

if first term and common ratio is given then we can find any term of Geometric progression with the help of formula t_{n}=ar^{n-1}

for example
if first term is 5 and common ratio is 4
then series will be 5,5*4,5*4*4,5*4*4*4……and so on

if we want to find 10th term then according to rule

t_{10}= a*r^{10-1}

=5*4^{10-1}=5*4^{9}

**How to find series and common ratio if first and last term of Geometric progression has given**

Let a be the first term and b be the last term
Then acording to formula of nth term nth term of g.p will be t_{n}=ar^{n-1}

t_{n}/a= r^{n-1}

Common ratio of G.P is
r=(t_{n}/a)^{1/n-1}

and series will be a,ar,ar^{2}

means a,a*(t_{n}/a)^{1/n-1}, a*(t_{n}/a)^{2/n-1}, a*(t_{n}/a)^{3/n-1} and so on