Question

Find sum.
$3\times 8+6\times 11+9\times 14+....$

Answer 1

The given series is $3\times 8+6\times 11+9\times 14+...$

$a_n=(n^{th}termof3,6,9,...)\times (n^{th}termof8,11,14,...)$

$=3n(3n+5)$

$\therefore S_n={\sum }_{k=1}^n{a}_k$

$={\sum }_{k=1}^n(9k^2+15k)$

$=9{\sum }_{k=1}^n{k}^2+15{\sum }_{k=1}^{n}k$

$=9\times \frac{n(n+1)(2n+1)}{6}+15\times \frac{n(n+1)}{2}$

$=\frac{3n(n+1)}{2}(2n+1+5)$

$=\frac{3n(n+1)}{2}(2n+6)$

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