Please explain the concept of finding the sum of an infinite Geometric Progression.

Hello student,
First of all a geometric progression or a geometric series is a series of terms in which the ratio of any two adjacent terms is constant.
Eg. 2,,4 8, ….. is a geometric progression where the common ratio i.e. r = 2.
Now we discuss how to find the sum of infinite Geometric Progression:
Firstly, the sum of n terms of a G.P. is given by
S = a(1-rⁿ)/(1-r), where r ≠ 1.
Now, if |r| < 1 and n → ∞ then rⁿ → 0 and in this case geometric series will be summable upto infinity and its sum is given by S_∞= a/(1-r).
I’ll consider one example here based on this concept:

Eg: The sum of an infinite number of terms of a G.P. is 15 and the sum of their squaresis 45. Find the series.
Sol: We have a + ar + ar² + ar³ + ....... = 15
so, a/(1-r) = 15.
a² + (ar)² + (ar²)² +.... = 45



(1-r)/(1+r) = 1/5
Solving this we get 5-5r = 1+r.
So r = 2/3.
so, a = 15(1-r) = 15(1-2/3) = 5.
so the series is 5, 10/3, 20/9, ….. .