Induction and Recursion.. I have NO idea!

  1. I think I'm pretty good at standard induction. Never had a problem.

    Induction and recursion is mercilessly whooping my ***.

    1. The problem statement, all variables and given/known data

    Let a1 = 1. For each natural number n > 1, let an = 3an-1 - 1.

    Prove that for each natural number n, an = 1/2(3n-1 + 1)

    2. Relevant equations



    3. The attempt at a solution

    WAT

    I can "build it." And I don't even feel comfortable doing that. Let S be the set of n for which the theorem holds.

    Let n = 2

    Theorem holds.

    Let n = 3

    Theorem holds.

    So 2,3 are elements of S.

    (Do I have to show it for 2? Or more? Or just one? Why?)

    Suppose that n >= 3 and {1,2,....,n} is a subset of S.

    I have no idea what to do. At all. Can someone please explain recursion induction to me?
     
  2. jcsd
  3. I think I see..

    I know that
    an+1 = 3an - 1 because that's the definition.

    And based on my assumption, an = 1/2(3n-1[/SUP + 1)

    So I can plug it in, and algebraically show that an+1 equals 1/2(3n[/SUP + 1)

    Am I on the right track?
     
  4. micromass

    micromass 18,448
    Staff Emeritus
    Science Advisor

    Yes, that's the idea. Can you do that?
     
  5. Yes, that's what I tried and it worked out.

    So for this example, I'd really only need to show it to be true for one value, right?
     
  6. micromass

    micromass 18,448
    Staff Emeritus
    Science Advisor

    Yes, you only need to show it's true for n=1 and you need to show the induction step.
     
  7. Do you mean n = 2? If n = 1, an-1 is undefined?
     
  8. micromass

    micromass 18,448
    Staff Emeritus
    Science Advisor

    Yes, check it for n=2 of course.
     
  9. Ok, got it. Thanks again!
     
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