Chapter F, Problem 17E

To determine

To write: The sum $1+2+4+8+16+32$ in sigma notation.

Answer to Problem 17E

The sum $1+2+4+8+16+32$ expressed in sigma notation as ${\displaystyle \sum _{i=0}^5{2}^i}$.

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:

Here, the sum $1+2+4+8+16+32$ is the sum of 6 numbers and number is of the form $2^i$, where $i$ varies from 0 to 5.

Thus, by the above definition the sum $1+2+4+8+16+32$ expressed as follows.

$1+2+4+8+16+32={\displaystyle \sum _{i=0}^5{2}^i}$

Thus, the sum $1+2+4+8+16+32$ expressed in sigma notation as ${\displaystyle \sum _{i=0}^5{2}^i}$.