Question

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

Answer 1

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

By observation, the first & second parts of

each term form an A.P

The $n^{th}$ term would be $\Rightarrow \left(3n\right)\times (3n+5)$

${\sum }_{n=1}^n\left(3n\right)(3n+5)$

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

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

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

$\Rightarrow \frac{3n(n+1)[2n+1+5]}{2}$

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