# -1^n = 1 or -1

• Sep 5th 2012, 07:59 AM
kasper90
(-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 ?
• Sep 5th 2012, 08:01 AM
Prove It
Re: -1^n = 1 or -1
Quote:

Originally Posted by kasper90
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 ?

-1^n = -1, since you need to do the exponentiation before the negation.
• Sep 5th 2012, 08:18 AM
kasper90
Re: (-1)^n = 1 or -1
I actually meant to say (-1)^n.
sorry for that
• Sep 5th 2012, 08:20 AM
Prove It
Re: -1^n = 1 or -1
(-1)^n = 1 if n is even, and (-1)^n = -1 if n is odd.
• Sep 5th 2012, 08:22 AM
kasper90
Re: -1^n = 1 or -1
Okay, but how do I prove that ?
• Sep 5th 2012, 08:32 AM
Plato
Re: -1^n = 1 or -1
Quote:

Originally Posted by kasper90
Okay, but how do I prove that ?

If $n$ is even, $n=2k$ then $(-1)^n=(-1)^{2k}=[(-1)^{2}]^k=[1]^k=1$.

If $n$ is odd, $n=2k+1$ then (WHAT?)
• Sep 5th 2012, 08:34 AM
kasper90
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.
• Sep 5th 2012, 08:37 AM
kasper90
Re: -1^n = 1 or -1
Plato posted his post just before I did,
but I think I got the idea, right ?

Is this a valid proof ?
• Sep 5th 2012, 08:51 AM
Plato
Re: -1^n = 1 or -1
Quote:

Originally Posted by kasper90
Plato posted his post just before I did,
but I think I got the idea, right ? Is this a valid proof ?

You have proved that $(-1)^{2k}=1$ so use it.
$(-1)^{2k+1}=(-1)^{2k}(-1)^{1}=(1)(-1)=-1$.