Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

johnny0929

  • one year 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

  • This Question is Closed
  1. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    something is wrong with that formula

  2. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    what is wrong with the formula?

  3. KingGeorge
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Looks correct to me.

  4. KingGeorge
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    That make sense?

  9. johnny0929
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  10. KingGeorge
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    You're welcome.

  11. satellite73
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    oh i read the exponent wrong, sorry

  12. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.