Question

Sum to $n$ terms the series :
$2\times 5+5\times 8+8\times 11+11\times 14+..$

Answer 1

$S=2\times 5+5\times 8+8\times 11+11\times 14+...n$ terms

We have $T_n=a+(n-1)d$

$T_n=[2+(n-1)3][5+(n-1)3]$

$=[2+3n-3][5+3n-3]$

$=(3n-1)(3n+2)$

$=9n^2+6n-3n-2$

$=9n^2+3n-2$

$S_n=9\sum n^2+3\sum n-2\sum 1$

$=9\left(\frac{n(n+1)(n+2)}{6}\right)+3\frac{n(n+1)}{2}-2n$

$=\frac{3n(n+1)(2n+1)}{2}+\frac{3n(n+1)}{2}-2n$

$=\frac{3n(n+1)}{2}[2n+1-1]-2n$

$=\frac{3n(n+1)}{2}\left[2(n+1)\right]-2n$

$=3n{(n+1)}^2-2n$

$=n[3{(n+1)}^2-2]$

$\therefore S_n=n[3n^2+6n+1]$