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  • If , for <img alt="k=1,2,3,4, \ldots \ldots, n," src="https://e

If a_k=\frac{1}{k(k+1)}, for k=1,2,3,4, \ldots \ldots, n,  then \left(\sum_{k=1}^n a_k\right)^2=

Option: 1

\left(\frac{n}{n+1}\right)


Option: 2

\left(\frac{n}{n+1}\right)^2


Option: 3

\left(\frac{n}{n+1}\right)^4


Option: 4

\left(\frac{n}{n+1}\right)^6


Answers (1)

best_answer

\begin{aligned} & \sum_{k=1}^n a_k=\sum_{k=1}^n \frac{1}{k(k+1)} \\ = & \left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\ldots+\left(\frac{1}{n}-\frac{1}{n+1}\right) \\ = & 1-\frac{1}{n+1}=\frac{n}{n+1} \\ & \left(\sum_{k=1}^n a_k\right)^2=\left(\frac{n}{n+1}\right)^2 \end{aligned}

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shivangi.bhatnagar

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