2(-1)^n = -2? Problem with (-1) to the power of natural numbers

[Joe123]

EDIT: Found the answer, seems I overlooked part of the solution in the learning materials ( answer extended into another page) the Solution does indeed equal what i thought it did.

Homework Statement

So this is the problem i have:

(2(-1)^n -((n*pi)^2(-1)^n)-2)*(8/(n*pi)^3)

where n = any natural number

this equation equals:

( (n*pi)^2-4)*(8/(n*pi)^3)

i cannot figure out how this works, if n = 1 the equation works, but if n = 2, then shouldn't the answer be:

( -(n*pi)^2)*(8/(n*pi)^3)

?

for simplification, removing the (8/(n*pi)^3)

(2(-1)^n -((n*pi)^2(-1)^n)-2)

where n = any natural number

=

( (n*pi)^2-4)

or

( -(n*pi)^2)

?


Any help would be appreciated.

Homework Equations

The Attempt at a Solution

See above.

 

[HallsofIvy]

If n is even then (-1)^n= 1. If n is odd, (-1)^n= -1. So
1) if n is even, 2(-1)^n -((n*pi)^2(-1)^n)-2)= 2- (n^2 pi^2- 2)= 4- n^2 pi^2

2) if n is odd, 2(-1)^n -((n*pi)^2(-1)^n)-2)= -2-(-n^2 pi^2- 2)= n^2 pi^2