Question

Find the sum to $n$ terms of the sequence $<a_n>$ given by $a_n=2^n+3n,nϵN$.

• A. $2(2^n+1)+\frac{3n}{2}(n+1)$
• B. $2(2^n-1)+\frac{3n}{2}(n+1)$
• C. $(2^n-1)+\frac{3n}{2}(n+1)$
• D. None of these

Answer 1

The correct option is B $2(2^n-1)+\frac{3n}{2}(n+1)$

Let $S_n=\sum a_n$
$=a_1+a_2+....+a_n$
$=(2^1+3\times 1)+(2^2+3\times 2)+(2^3+3\times 3+....+(2^n+3\times n)$
$=(2^1+2^2+2^3+...+2^n)+3(1+2+3+...+n)$
$=2\left(\frac{2^n-1}{2-1}\right)+3\left[\frac{n}{2}\right(n+1\left)\right]$
$=2(2^n-1)+\frac{3n}{2}(n+1)$.