Explanation

(a)
Given series is ${1.3}^{2} + {2.5}^{2} + {3.7}^{2} + ...\infty$
This is an arithmetic-geometric series whose $n$th term is equal to
$T_{n} = n(2n + 1)^{2} = 4n^{3} + 4n^{2} + n$
$\therefore S_{n} = \sum_{1}^{n}T_{n} = \sum_{1}^{n}{(4n^{3} + 4n^{2} + n)}$
$= 4\sum_{1}^{n}n^{3} + 4\sum_{1}^{n}n^{2} + \sum_{1}^{n}n$
$= 4\left( \frac{n}{2}(n + 1) \right)^{2} + \frac{4}{6}n(n + 1)(2n + 1) + \frac{n}{2}(n + 1)$
$= n(n + 1)\left\lbrack n^{2} + n + \frac{4}{6}(2n + 1) + \frac{1}{2} \right\rbrack$
$= \frac{n}{6}(n + 1)(6n^{2} + 14n + 7)$