Question

Find the sum of first $x$ terms of the series:
$3+7+13+21+31+....................$

Answer 1

The given series (upto $x$ terms) can be written as

$(1+2)+(1+2+4)+(1+2+4+6)+.......+(1+2+4+....+2x)$.

The $n^{th}$ term of the series is $t_n=(1+2+4+.....+2n)=1+2(1+2+...n)=1+n+n^2$.

So, the sum of first $x$ terms of the given series is

$\sum _{n=1}^x(1+n+n^2)$

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

$=\frac{x(x^2+3x+5)}{3}$.