Chapter F, Problem 4E

Write the sum in expanded form.

4. ${\displaystyle \underset{i=4}{\overset{6}{\sum }}{i}^3}$

To determine

To write: The sum ${\displaystyle \sum _{i=4}^6{i}^3}$ is in expanded form.

Answer to Problem 4E

The expanded form of the sum ${\displaystyle \sum _{i=4}^6{i}^3}$ is $4^3+5^3+6^3$.

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 expression ${\displaystyle \sum _{i=4}^6{i}^3}$ simplified as follows.

${\displaystyle \sum _{i=4}^6{i}^3}=4^3+5^3+6^3$

Thus, the expanded form of the sum ${\displaystyle \sum _{i=4}^6{i}^3}$ is $4^3+5^3+6^3$.