Question

Find the sum of $n$ term of the series $3\times 8+6\times 11+9\times 14+....$

Answer 1

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

$a_n=(n^{th}$ term of $3,6,9,...)\times$({n}^{th}$termof$8,11,14,...)$

$=\left(3n\right)(3n+5)$

$=9n^2+15n$

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

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

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

$=\frac{3n(n+1)(2n+1)}{2}+\frac{15n(n+1)}{2}$

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

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

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