Question

Find the $n^{th}$ term and sum to $n$ terms of the following series:
$3+6+11+18+$____

Answer 1

$S_n=3+6+11+\mathrm{18....}+n^{th}$ term

$S_n=(1^2+2)+(2^2+2)+(3^2+2)+(4^2+2).......+(n^2+2)$

As we can see that the general term is $n^2+2$

So, the $n^{th}$ term is $n^2+2$

Sum of given series upto $n$ terms:

$S_n=(1^2+2^2+3^2+4^2......+n^2)+2n$

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

$S_{n}n(2n^2+3n+13)/6$