Question

Find the $n^{th}$ term and the sum to $n$ terms of the following series: $2+5+13+35+....$

Answer 1

The series of numbers is $3,5,13,35$

$3=2^0+3^0$

$5=2^1+3^1$

$13=2^2+3^2$

$35=2^3+3^3$

Hence general term is $2^k+3^k$.

Sum of the terms $={\sum }_{k=0}^n{2}^k+3^k=\frac{2^n-1}{2-1}+\frac{3^n-1}{3-1}=2^n-1+\frac{1}{2}(3^n-1)$