Question

Find $n^{th}$ term and sum of $n^{th}$ term of the sequence $2,5,12,31,86,\dots n.$

Answer 1

$2,5,12,31,86,......n$

first differences $=3,7,19,\mathrm{55......}$

second differences $4,12,\mathrm{36......}$

which is a group

$\therefore$ $n^{th}$ term is of the form $4\left(n\right)={ar}^{n-1}+b^n+c$

substituting $n=1,2,3$ & $r=3$, we get

$a=1$, $b=1$, $c=0$

$\therefore$ general term $=3^{n-1}+3$

sum $=\sum 3^{n-1}+\sum n$

$=\frac{3^n+n^2+n-1}{2}$