Chapter F, Problem 27E

To determine

To find: The value of the sum ${\displaystyle \sum _{i=0}^4\left(2^i+i^2\right)}$.

Answer to Problem 27E

The value of the sum ${\displaystyle \sum _{i=0}^4\left(2^i+i^2\right)}$ is $61$.

Explanation of Solution

Definition used:

If $a_m,a_{m+1},\mathrm{...},a_n$ are real numbers and m and n are integers such that $m\le n$, then ${\displaystyle \sum _{i=m}^n{a}_i=a_m+}{a}_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_n$.

Calculation:

By the above definition, the sum ${\displaystyle \sum _{i=0}^4\left(2^i+i^2\right)}$ expressed as follows.

$\begin{array}{c}{\displaystyle \sum _{i=0}^4\left(2^i+i^2\right)}=\left(2^0+0^2\right)+\left(2^1+1^2\right)+\left(2^2+2^2\right)+\left(2^3+3^2\right)+\left(2^4+4^2\right)\\ =\left(1+0\right)+\left(2+1\right)+\left(4+4\right)+\left(8+9\right)+\left(16+16\right)\\ =1+3+8+17+32\\ =61\end{array}$

Thus value of the sum ${\displaystyle \sum _{i=0}^4\left(2^i+i^2\right)}$ is $61$.