johnny0929 2 years ago 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

1. satellite73

something is wrong with that formula

2. johnny0929

what is wrong with the formula?

3. KingGeorge

Looks correct to me.

4. KingGeorge

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.

5. johnny0929

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

6. KingGeorge

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

7. KingGeorge

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 $(x-1)x^n+(x^n-1)=x^{n+1}-x^n+x^n-1=x^{n+1}-1.$

8. KingGeorge

That make sense?

9. johnny0929

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

10. KingGeorge

You're welcome.

11. satellite73

oh i read the exponent wrong, sorry