Mathematical Induction question (1 Viewer)

[Dream Theater]

Hey, I have to prove that x^n + 1 is divisible by x + 1

this is an image of what I'm doing:

I get stuck where I'm trying to make that 1 equal to x, factorise and put my result for x^k + 1 back in.

Thanks in advance for the help.

 

[CM_Tutor]

The problem is in the question - I think you will find that the correct theorem is that xⁿ + 1 is divisible by x + 1 for odd positive integers n, not for all positive integers n.

 

[:: ck ::]

in which case you would do n=k+2 instead?

its been a yr since i last did induction >.<!!

 

[hatty]

Originally posted by CM_Tutor
The problem is in the question - I think you will find that the correct theorem is that xⁿ + 1 is divisible by x + 1 for odd positive integers n, not for all positive integers n.

what a genius

 

[CM_Tutor]

Originally posted by :: ryan.cck ::
in which case you would do n=k+2 instead?

its been a yr since i last did induction >.<!!

That's one alternative - you can make your assumption 'let k be an odd positive value of n'. Then look at n = k + 2.

Or, you can restate the theorem. Put n = 2m - 1, so that the odd positive integers n = 1, 3, 5, 7, ... correspond to the positive integers m = 1, 2, 3, 4, ...

So, the theorem becomes that x^2m-1 + 1 is divisible by x + 1 for all positive integers m, and do the induction as normal.

 

[:: ck ::]

wahh so smart!! >.<"

thxthx