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Random Reddit question 3

Random Reddit question 3

The question posed is as follows:

Let $S_n$ be defined by

$$S_n = \frac{n(n+1)}{2(2n+1)}$$

By induction, prove that

$$S_n = \frac{1^2}{1\times 3} + \frac{2^2}{3\times 5} + \frac{3^2}{5\times 7}+\dots +\frac{n^2}{(2n-1)(2n+1)} $$

 

Nice proof by induction - the tricky step is the factorisation!

sketch of proof below