Question

Find the sum of n terms off the series $2+5+12+31+86+.....$

Answer 1

The above series can be written as

$(3^0+1)+(3^1+2)+(3^2+3)+(3^3+4)+....$

Therefore

$T_n=3^n+(n+1)$

$\sum (3^n+n+1)$

$={\sum }_{n=0}^n{3}^n+{\sum }_{n=0}^n(n+1)$

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

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