Question

Explain why$\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}i{j}^2{k}^3=\sum _{j=1}^m\sum _{k=1}^n\sum _{i=1}^{l}i{j}^2{k}^3$.

Short answer

The two summations $\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}i{j}^2{k}^3=\sum _{j=1}^m\sum _{k=1}^n\sum _{i=1}^{l}i{j}^2{k}^3$contains the same summands although the summands are rearranged by the cumulative and associative properties of addition, they are equal.

Step-by-step solution

Step 1. Given information

We need to explain why$\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}i{j}^2{k}^3=\sum _{j=1}^m\sum _{k=1}^n\sum _{i=1}^{l}i{j}^2{k}^3$.

Step 2. Definition for the summation:

The summation definition states that if the order of the summation will change but there is no effect on the result. The two summations $\sum _{i=1}^l\sum _{j=1}^m\sum _{k=1}^{n}i{j}^2{k}^3=\sum _{j=1}^m\sum _{k=1}^n\sum _{i=1}^{l}i{j}^2{k}^3$contains the same summands although the summands are rearranged by the cumulative and associative properties of addition, they are equal.