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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 byS = 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 + ....... = 15so, a/(1-r) = 15.a2 + (ar)2 + (ar2)2 +.... = 45(1-r)/(1+r) = 1/5Solving 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, ….. .
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