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Help with induction

  1. Jan 14, 2009 #1
    Use mathematical induction to prove

    1*3^1 + 2*3^2 + 3*3^3 + ... + n*3^n = 1/4(2n -1)3^n+1 +3/4

    Let p(n) be variable proposition 1/4(2n -1)3^n+1 + 3/4

    Is p(1) true?

    1/4(2*1 - 1)3^1+1 + 3/4 = 3 & 1*3^1 = 3 so p(1) is true.

    Now assume p(k) is true, that is 1/4(2k - 1)3^k+1 + 3/4 is true.

    Now prove p(k + 1) is true.

    If p(k + 1) is true we should have 1/4(2(k+1) - 1)3^(k+1) +1 + 3/4
    which is

    1/4(2k + 1)3^k+2 +3/4

    Now to proof.

    We have to prove that p(k) + p(k+1) = 1/4(2k + 1)3^k+2 + 3/4

    1/4(2k + 1)3^k+1 + 3/4 + (K+1)3^(k+1)

    Multiplying by 4

    (2k + 1)3^k+1 + 3 + 4(k+1)3^(k+1)

    This is where I get bogged down? Can anyone help please!!
     
  2. jcsd
  3. Jan 14, 2009 #2

    danago

    User Avatar
    Gold Member

    Should that be 2k - 1?

    Anyway, after fixing that, try taking out a common factor of 3^(k+1). See if you can go from there.
     
  4. Jan 14, 2009 #3
    Yeah you're right!! lol

    After some head scratching and crossings out I think I've cracked it!

    Anyway after fixing my error it simplifies to

    18k*3^k + 9*3^k +3

    (2k + 1)9*3^k + 3

    But 9 = 3^2 so we have

    (2k + 1)3^2*3^k + 3
    (2k + 1)3^k+2 +3

    Now multiply by 1/4 and there is the answer!!

    1/4(2k + 1)3^k+2 +3/4


    I think!!

    Many thanks for your help.

    James
     
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