The sums of the following expressions are true for any positive integer n. 

 

1.      .

 

Example:

            Let n = 3, the left-hand side of the above expression is 1-3+5 = 3 and its right-hand side is –3(-1) = 3. 

 

 

2.      .

 

Example:

            Let n = 3, the left-hand side of the above expression is 1-9+25 = 17 and its right-hand side is –2(9)(-1) –1 = 17. 

 

 

3.      .

 

Example:

            Let n = 4, the left-hand side of the above expression is 1-27+125-343 = -244 and its right-hand side is [3(4) – 4(64)](1) = -244. 

 

 

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In-Text or Website Citation

Tue N. Vu, Finite Alternative Odd Power Sum, 10/15/2006, from Series Math Study Resource.
Hyperlink: http://www.seriesmathstudy.com/finitealternativeodd.