<s>[INST] Prove the sum to n terms of the series Prove that the sum to $n$ terms of a sequence:
$$\frac{1^2}{1\times 3}+\frac{2^2}{3\times 5}+\frac{3^2}{5\times 7}+\cdots $$
is 
$$
\frac{n(n+1)}{2(2n+1)}
$$
 [/INST] Assuming that $n^2, (2n-1), (2n+1)$ are the general terms for $\{1^2, 2^2, 3^3, \cdots, \}$, $\{1, 3, 5,  \cdots, \}$, $\{3, 5, 7 \cdots, \}$ we see that $t_n=\frac{n^2}{(2n-1)(2n+1)}$ which partial fraction decomposition gives as $\frac 18[\frac{1}{2n-1}-\frac{1}{2n+1}]$. take the sum over $n$ from 1 to $n$ and get desired answer.
</s>