Question

find the sum of 1.3+2.5+3.7+...n(2n+1)

Answer 1

This is equivalent to
Summation of
$n(2n + 1) \\ = 2 \times s {n}^{2} + sn$
Here s means the summation symbol.
And we already know that
$s {n}^{2} = \frac{n(n + 1)(2n + 1)}{6}$
and
$sn = \frac{n(n + 1)}{2}$
Now we substitute these values to get the answer.
$x = 2 \times \frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2} \\ x = \frac{n(n + 1)(2n + 1)}{3} + \frac{n(n + 1)}{2} \\6 x = 2n(n + 1)(2n + 1) + 3n(n + 1) \\ 6x = n(n + 1)(4 {n}^{2} + 2 + 3) \\ x = \frac{n(n + 1)(4 {n}^{2} + 5)}{6}$