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- September 5th 2012, 07:59 AM #1

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## (-1)^n = 1 or -1

Okay, this might be a little bit stupid question...

I'm studying ross elementary analysis.

And I'm trying to get some skills in writing formal proofs, instead of intuitive ideas.

I want to proof that the sequence (-1)^n = 1 or -1.

But how do I give a real proof for this ?

- September 5th 2012, 08:01 AM #2

- September 5th 2012, 08:18 AM #3

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- September 5th 2012, 08:20 AM #4

- September 5th 2012, 08:22 AM #5

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- September 5th 2012, 08:32 AM #6

- September 5th 2012, 08:34 AM #7

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## Re: -1^n = 1 or -1

Okay, I'll try:

If n is even,

than ∃k∈N : n = 2k,

than (-1)^n = (-1)^2k=[(-1)^2]^k = 1^k

If n is uneven,

than ∃k∈N : n=2k-1,

than (-1)^n = (-1)^[2k-1] = -1^k,

Induction (prove that 1^k =1)

1^1

if 1^k = 1, than 1^(k+1)=1^k * 1 = 1^k = 1

By induction, 1^k =1 for all natural numbers.

- September 5th 2012, 08:37 AM #8

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- September 5th 2012, 08:51 AM #9