<s>[INST] Find the sum to $n$ terms of the series: $1^2.1+2^2.3+3^2.5+....$ Find the sum to $n$ terms of the series:
$$1^2.1+2^2.3+3^2.5+.....$$
My Attempt:
Here, $n^{th}$ term of $1,2,3,....=n$
$n^{th}$ term of $1^2,2^2,3^2,....=n^2$
Also, $n^{th}$ term of $1,3,5,....=2n-1$
Hence, $n^{th}$ term of the given series is $t_n=n^2(2n-1)$
 [/INST] $$\sum_{k=1}^nk^2(2k-1)=\sum_{k=1}^n\left[12\binom{k}3+10\binom{k}2+\binom{k}1\right]=12\binom{n+1}4+10\binom{n+1}3+\binom{n+1}2$$
This under the convention that $\binom{k}{r}=0$ if $r\notin\{0,\dots,k\}$.
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