Find the sum to n terms of each of the series in Exercises 1 to 7.

The given series is $3 \times 1^{2}+5 \times 2^{2}+7 \times 3^{2}+\ldots$

$n^{\text {th }}$ term, $a_{n}=(2 n+1) n^{2}=2 n^{3}+n^{2}$

$\therefore S_{n}=\sum_{k=1}^{n} a_{k}$

$=\sum_{k=1}^{n}=\left(2 k^{3}+k^{2}\right)=2 \sum_{k=1}^{n} k^{3}+\sum_{k=1}^{n} k^{2}$

$=2\left[\frac{n(n+1)}{2}\right]^{2}+\frac{n(n+1)(2 n+1)}{6}$

$=\frac{n^{2}(n+1)}{2}+\frac{n(n+1)(2 n+1)}{6}$

$=\frac{n(n+1)}{2}\left[n(n+1)+\frac{2 n+1}{3}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3 n^{2}+3 n+2 n+1}{3}\right]$

$=\frac{n(n+1)}{2}\left[\frac{3 n^{2}+5 n+1}{3}\right]$

$=\frac{n(n+1)\left(3 n^{2}+5 n+1\right)}{6}$