algebra precalculus - Find the $n^{th}$ term and the sum to $n$ terms of the following series

Find the $n^{th}$ term and sum to $n$ terms of the following series.

$$1.3+2.4+3.5+……$$

My Attempt:

Here,
$n^{th}$ term of $1+2+3+……=n$

$n^{th}$ term of $3+4+5+……=n+2$

Thus,

$n^{th}$ term of the series $1.3+2.4+3.5+……=n(n+2)$

$$t_n=n^2+2n$$

If $S_n$ be the sum to $n$ terms of the series then

$$S_n=\sum t_n$$
$$=\sum (n^2+2n)$$

How do I proceed?

Answer

$$\begin{align}\sum_{i=1}^n (i^2+2i) &=\sum_{i=1}^n i^2 + 2\sum_{i=1}^n i\\ &= \frac{n(n+1)(2n+1)}{6}+2\cdot \frac{n(n+1)}{2} \end{align}$$

You might want to factorize the terms to simplify things.