(x^k) - 1 = (x - 1)*(x^(k-1) + x^(k-2) + + x + 1)

[General_Sax]

(x^k) - 1 = (x - 1)*(x^(k-1) + x^(k-2) + ... + x + 1)

Where does this factorization come from? I need to know so I can use it in a proof. Thanks.

 

[micromass]

Work out the right side...

 

[Dickfore]

Use mathematical induction:
<br /> x^{k+1} - 1 = (x^{k+1} - x^k) + (x^k - 1) = (x - 1) x^k +(x^k - 1)<br />

 

[General_Sax]

Work out the right side...

So, there is no theorem to use?

@Dickfore

I'm confused as to the next step -- yes I've been trying to work it out.

Should I try to factor (x-1) out of the expression?

 

[Dickfore]

@Dickfore

I'm confused as to the next step -- yes I've been trying to work it out.

Should I try to factor (x-1) out of the expression?

According to P.M.I., you should substitute the expression that you're trying to prove for n = k, in this case for x^k - 1, factor x - 1, and see if you get the corresponding expression for n = k + 1. If you do, then the proof is complete.

 

[epenguin]

So, there is no theorem to use?

Could almost say this is the theorem.

Multiply your last bracket by x
on the next line multiply that same last bracket by -1.
Compare. Add.

This is a very useful formula.
That last bracket is the geometrical series for example, hence you calculate it equals (x^k - 1)/(x - 1). Heard anything like that before?

 

[General_Sax]

Thanks for the help people. I think I've got it. Just used formula for geometric series and did some algebra -- hope it's good enough.

 

[Fredrik]

Thanks for the help people. I think I've got it. Just used formula for geometric series and did some algebra -- hope it's good enough.

Did you really mean that you used the formula for the geometric series? This theorem?

For all real numbers x such that |x|<1, $\sum_{k=0}^\infty x^k=\frac{1}{1-x}.$​

This theorem is much harder to understand than the formula you're trying to prove, because it involves convergence. Its standard proof uses the formula you're trying to prove, which by the way holds for all real numbers x, not just the ones that satisfy |x|<1.

So, there is no theorem to use?

Only the distributive laws a(b+c)=ab+ac, (a+b)c=ac+bc, and if you want to do a rigorous proof that the result of the multiplication is what you (should) think it is, you have to use induction.

Should I try to factor (x-1) out of the expression?

Wouldn't that give you two factors of (x-1)? That would only make things worse.

 

[General_Sax]

Thanks for the additional effort/attention Fredrik, but I've already used the geometric "series" -- perhaps it's more accurate to use the term "sum" -- in a proof.

http://en.wikipedia.org/wiki/Geometric_series#Formula

that's the one I used. Just split it up w/ some algebra. It's for a CMPUT course and we haven't much experience with proofs so I don't think they expect much rigour.

 

[Fredrik]

So basically you took the formula $1+x+\cdots+x^{k-1}=\frac{1-x^{k}}{1-x}$ (which doesn't hold for x=1 by the way) and just multiplied it by x-1? If I was your teacher, I wouldn't accept that as an answer. (The formula you found is too similar to what you're supposed to prove). Why don't you just do the multiplication on your right-hand side to see what you get? Do you know how to evaluate a(b+c)? How about (a+b)(b+c)? How about (a+b)(b+c+d)?