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Question

If Sn denote the sum of n terms of a G.P. whose first term is a,andcommonratior,findthesumofS_{1}, S_{3}, S_{5}, ...S_{2n - 1}$.

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Solution

Let Sn be the sum of first n terms of a G.P. and r be common ratio
Formula for sum of n terms of G.P. is given by
Sn=a(rn1)r1
Then, S1=a(r11)r1

S3=a(r31)r1
S5=a(r51)r1
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S2n1=a(r2n11)r1
Adding all terms, we get
S=ar1[(r+r3+.....+r2n1)+(1)(n)]
Now, r,r3,r5,....,r2n1 represents G.P. with common ration r2 and first term r
S=ar1[r(r2n1)r21n]

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