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  4. Let Sk be sum of an infinite G.P. whose first term is k and common ratio is (1/k+1). Then displaystyle∑k=110 Sk is equal to

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Q. Let $S_{k}$ be sum of an infinite $G.P$. whose first term is $k$ and common ratio is $\frac{1}{k+1}$. Then $\displaystyle\sum_{k=1}^{10} S_{k}$ is equal to ____

Sequences and Series

Solution:

$S_{k}=\frac{k}{1-\frac{1}{k+1}}=k+1$
$ \displaystyle\sum_{k=1}^{10} S_{k}=\displaystyle\sum_{k=1}^{10}(k+1)=\frac{10 \times 11}{2}+10=65$