1. Not finding help here? Sign up for a free 30min 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!

Help needed with a likely rather obvious induction proof

  1. Dec 29, 2006 #1
    1. The problem statement, all variables and given/known data
    show that for all natural numbers n: 3^(2n+1)+2^(n-1) is divisible by 7

    3. The attempt at a solution

    i've been trying to get the second part of the proof to look like the first part, so as to be able to conclude some multiple is also divisible by 7, but i don't seem to get what needs to be done..
    3^(2(n+1)+1)+2^(n+1-1) -> 3^(2n+2+1)+2^(n+1-1) -> 3²*3^(2n+1)+2*2^(n-1) (= 7*k)
    only here i sort of get stuck trying to get the multipliers out, and i'm not certain enough of my math 'certain knowledge' otherwise to just posit that 3*(something)+2*(something) always yields multiples of 7 (not that it does, in this case)

    am i really trying to go down the wrong path here? or am i just missing something entirely too obvious? :(
  2. jcsd
  3. Dec 29, 2006 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, as you say, 3*(something)+ 2*(something) does NOT always yield multiples of 7 (for example if each "something" is 1, the sum is 5 which is not a multiple of 5 so you cannot "posit" that it always does!

    What you need to show is that if 3^(2N+1)+2^(N-1) is a multiple of 7 for some specific N (do you see the difference between that and "3^(2n+1)+2^(n-1) is a multiple of 7 for all n?) then 3^(2(N+1)+1)+2^((N+1)-1) is also a multiple of 7.

    3^(2(N+1)+1)+2^(N+1-1)= 3^(2N+1+2)+2^(N+1-1)= 3²*3^(2N+1)+2*2^(N-1)= 9(3^(2N+1))+2(2^(N-1))= 2[3^(2n+1)+ 2^(N-1)]+ 7(3^(2N+1). Now, you know that 3^(2N+1)+ 2^(N-1) is a multiple of 7: 3^(2N+1)- 2^(N-1)= 7m. What does that tell you about 2[3^(2n+1)+ 2^(N-1)]+ 7(3^(2N+1)?
  4. Dec 29, 2006 #3
    ugh.. as i suspected, totally obvious :(
    thank you for the quick reply, HallsofIvy :)
    Last edited by a moderator: Dec 29, 2006
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Help needed with a likely rather obvious induction proof
  1. Induction Proof Help (Replies: 3)