Question

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

Answer 1

$2,5,12,31,86$

the $n^{th}$ term of series $=3^{n-1}+n$

sum of $n^{th}term$=$\sum (3^{n-1}+n)$

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

$=(1+3+9+27+.....nterms)$

$+(1+2+3+\mathrm{4.....}+nterms)$

$=\frac{9(r^n-1)}{r-1}+\frac{n(n+1)}{2}$

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

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