Question

Find the sum of $n$ terms of the series $2.5+5.8+8.11+11.14+....$.

Answer 1

The First sequence in the series is $2+5+8+\dots \text{n terms}$

This is an A.P. with first term as $2$ and common difference $3$

$\therefore T_1=2+3(n-1)=3n-1$

The Second sequence in the series is $5+8+11\dots \text{n terms}$

This is an A.P. with first term as $5$ and common difference $3$

$\therefore T_2=5+3(n-1)=3n+2$

Thus $n^{th}$ term of the series $=(3n-1)(3n+2)=9n^2+3n-2$

Therefore, $S_n=\sum (9n^2+3n-2)=\frac{9n(n+1)(2n+1)}{6}+\frac{3n(n+1)}{2}-2n=3n^2(n+1)$