Question

Solve :
${1.2}^2+{2.3}^2+{3.4}^2+...$

Answer 1

$P={1.2}^2+{2.3}^2+{3.4}^2+.....$

general term $\Rightarrow n(n+1{)}^2$

$\therefore P=\sum _{n=1}^{n}n(n+1{)}^2$

$=\sum _{n=1}^n(n^3+n^2+n)$

$P=\frac{n^2(n+1{)}^2}{4}+2\times \frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}$

$\Rightarrow P=\frac{n(n+1)\left[n\right(n+1)\times 3+4(2n+1)+6]}{12}$

$\Rightarrow P=\frac{n(n+1)}{2}[3n^2+3n+8n+4+6]$

$P=\frac{n(n+1)(3n^2+11n+10)}{12}$.