Chapter E, Problem 39E

Prove formula (e) of Theorem 3 using a method similar to that of Example 5, Solution 1 [start with (1 + i)⁴ − i⁴].

3 Theorem Let c be a constant and n a positive integer. Then

1. (a) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}1=n}$
2. (b) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}c=nc}$
3. (c) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}i=\frac{n(n+1)}{2}}$
4. (d) ${\displaystyle \underset{i=1}{\overset{n}{\sum }}{i}^2=\frac{n(n+1)(2n+1)}{6}}$
5. (e) ${{\displaystyle \underset{i=1}{\overset{n}{\sum }}{i}^3=\left[\frac{n(n+1)}{2}\right]}}^2$