Problem 42 (difficulty: 3/10)

Prove that the following identity holds for all positive integer \(\displaystyle n\):

\(\displaystyle \frac{1}{1\cdot 3} +\frac{1}{3\cdot 5}+\ldots + \frac{1}{(2n-1)\cdot (2n+1)}=\frac{n}{2n+1}.\)

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