Question

Find the sum to n terms of the series $\left({\text{n}}^{\text{2}}\text{-}{\text{1}}^{\text{2}}\right)\text{+ 2}\left({\text{n}}^{\text{2}}\text{-}{\text{2}}^{\text{2}}\right)\text{+ 3}\left({\text{n}}^{\text{2}}\text{-}{\text{3}}^{\text{2}}\right)\text{+}...$

Answer 1

$(n^2-1^2)+2(n^2-2^2)+3(n^2-3^2)+....$

$\Rightarrow n^2+2n^2+3n^2+...\dots +n{n}^2-[{1.1}^2+{2.2}^2+{3.3}^2+...\dots +n.n^2]$

$\Rightarrow n[1+2+3+...\dots +n]-[1^3+2^3+3^3+...\dots +n^3]$

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

$\Rightarrow \frac{n(n+1)}{2}[n-\frac{n(n+1)}{2}]$

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

$\Rightarrow \frac{n(n+1)(n-n^2)}{4}$

$\Rightarrow \frac{n^2(n+1)(1-n)}{4}$

$\Rightarrow \frac{n^2(1-n^2)}{4}$.