1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Prove, using induction

  1. Mar 23, 2005 #1
    I was asked to prove, using induction, that 34n+2 + 26n+3 is divisible by 17. I tried to do it, but I couldn't get anywhere. Can someone give me a push in the right direction?

    Here's my attempt:
    f(k) = 34k+2 + 26k+3
    f(k+1) = 34k+6 + 26k+9
    And now, all I have to do is prove that f(k+1) - f(k) = 17m, but I couldn't do it.


    I don't really see why induction is necessary anyway. Here's my induction-free attempt:

    [tex]3^{4n+2} + 2^{6n+3} = 9^{2n+1} + 8^{2n+1} = (9+8) \sum^{2k+1}_{n=1} 9^{2k+1-n} \; 8^{n-1} = 17m[/tex]
     
  2. jcsd
  3. Mar 23, 2005 #2

    mathman

    User Avatar
    Science Advisor

    Your proof is incorrect - the factoring is wrong.

    Proof of theorem: f(k+1)=81*34k+2+64*26k+3
    =17*34k+2+64*f(k), where both terms in the sum are divisible by 17. You need also f(0)=17 to start with.
     
  4. Mar 24, 2005 #3
    Thanks.

    By the way, did you mean that my induction-free proof was wrong?
     
  5. Mar 24, 2005 #4

    mathman

    User Avatar
    Science Advisor

    Yes: 92n+1+82n+1 does not factor as you wrote it. You confused it with an expression with a minus sign.
     
  6. Mar 24, 2005 #5
    Ah yes, it's:
    a^n + b^n = (a+b)[a^(n-1) - a^(n-2)b + ... + b^(n-1)], correct?
     
  7. Mar 24, 2005 #6

    HallsofIvy

    User Avatar
    Science Advisor

    You mean like a^2+ b^2= (a+ b)(a-b )???

    Oops, I guess it's not correct.
     
  8. Mar 24, 2005 #7

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Nope,it the REALS,you can't factor [tex] a^{n}+b^{n} [/tex]...

    Daniel.
     
  9. Mar 24, 2005 #8
    Well, if n is odd you can, and his newest attempt at a factorization is correct. This makes his original proof right, with a small modification.

    [tex]n \in \mathbb{N} \Longrightarrow 3^{4n+2} + 2^{6n+3} = 9^{2n+1} + 8^{2n+1} = 17\sum_{k=0}^{2n}(-1)^{k} \ 9^k \ 8^{2n-k}[/tex]
     
    Last edited: Mar 24, 2005
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Prove, using induction
  1. Prove by Induction (Replies: 4)

Loading...