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Question

Find a formula for Sn, the sum of the first n terms of the geometric sequence.


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Solution

Find the sum of first n terms of the geometric sequence.

Consider the G.P, a,ar,ar2,ar3,......arn-1

Let the sum of n terms be Sn, first term be a and the common ratio of G.P. be r.

Then,Sn=a+ar+ar2......arn-1 …...…….(1)

For r=1, Sn=a+a+a+.....a=na

For r≠1,

Multiply both sides of Sn by r, we get,

rSn=ar+ar2+ar3......arn ………….(2)

Subtracting (2) from (1), we get,

⇒rSn-Sn=(ar+ar2+ar3......arn-1+arn)-(a+ar+ar2+ar3......arn-1)

⇒rSn-Sn=arn-a

⇒(r-1)Sn=a(rn-1)⇒Sn=a(rn-1)(r-1)

Hence, sum of n terms of the geometric sequence is Sn=a(rn-1)(r-1).


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