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Question
In a geometric series, the first term =a, common ratio =r. If Sn denotes the sum of the n terms and Un=n∑n=1Sn, then rSn+(1−r)Un equals to
In a geometric series, the first term =a, common ratio =r. If Sn denotes the sum of the n terms and Un=n∑n=1Sn, then rSn+(1−r)Un equals to
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Solution
The correct option is C na
We know Sn=a(rn−1)r−1
⇒Un=n∑n=1a(rn−1)r−1=ar−1n∑n=1(rn−1)
⇒Un=ax−1{r+r2+...+rn−n}
=ar−1{r(rn−1)r−1−n}
⇒(r−1)Un=ar(rn−1)r−1−an
⇒(r−1)Un=rSn−an
⇒rSn+(1−r)Un=an
We know Sn=a(rn−1)r−1
⇒Un=n∑n=1a(rn−1)r−1=ar−1n∑n=1(rn−1)
⇒Un=ax−1{r+r2+...+rn−n}
=ar−1{r(rn−1)r−1−n}
⇒(r−1)Un=ar(rn−1)r−1−an
⇒(r−1)Un=rSn−an
⇒rSn+(1−r)Un=an
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