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 1,2,2²,2³,2⁴…….(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

t₁=a
t₂=ar=ar^2-1
t₃=ar2=ar^3-1
t₄=ar3=ar^4-1
t₅=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

t₁₀= a*r^10-1

=5*4^10-1=5*4⁹

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,ar²

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