## johnny0929 Group Title 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 one year ago one year ago

1. satellite73 Group Title

something is wrong with that formula

2. johnny0929 Group Title

what is wrong with the formula?

3. KingGeorge Group Title

Looks correct to me.

4. KingGeorge Group Title

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 Group Title

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 Group Title

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 Group Title

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 Group Title

That make sense?

9. johnny0929 Group Title

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

10. KingGeorge Group Title

You're welcome.

11. satellite73 Group Title

oh i read the exponent wrong, sorry