Question

prove by induction that ((x^(k))-1)=(x-1)(x^(k-1)+x^(k-2)+...+x+1)
i got stuck when i have to prove k=n+1 from k=n being assumed true

Answer 1

something is wrong with that formula

Answer 2

what is wrong with the formula?

Answer 3

Looks correct to me.

Answer 4

Anyways, you have to start with a base case of k=1. Then, you have\[(x-1)\overset{\text{?}}{=} (x-1)(1)\]Which is true, so the base case is good.

Answer 5

yup i have completed the base case and i got stuck when i assume k=n is true and i want to prove k=n+1 from k=n

Answer 6

Now you assume true up to some \(n\in\mathbb{Z}^+\). So \[(x^n-1)=(x-1)(x^{n-1}+x^{n-2}+...+x+1)\]

Answer 7

Now, look at \[(x-1)(x^n+x^{n-1}+x^{n-2}+...+x+1).\]This is equal to \[(x-1)x^n+(x-1)(x^{n-1}+x^{n-2}+...+x+1)\]which by our inductive hypothesis, is equal to \[(x-1)x^n+(x^n-1)=x^{n+1}-x^n+x^n-1=x^{n+1}-1.\]

Answer 8

That make sense?

Answer 9

ok that makes a lot of sense. thank you so much!

Answer 10

You're welcome.

Answer 11

oh i read the exponent wrong, sorry