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Proof by Induction with Exponents

  1. Apr 23, 2013 #1
    1. The problem statement, all variables and given/known data

    By mathematical induction, prove that for n ≥ 1, 4/(7n - 3n).



    2. Relevant equations



    3. The attempt at a solution

    I got the base case down P(1): 7-3=4.

    Now the actual problem,

    7n - 3n = 4x
    7n+1 - 3n+1 = 7(7n) - 3(3n)
    =7(4x + 3n) - 3(7n - 4x)
    =21x+ (7(3n)) - (3(7n)) + 12x

    -This is the point at which I get stuck there is nothing I can really factor out and I'm pretty sure I messed up earlier or there is something I have to do with the 7 and 3n. Any help would be appreciated.

    Trying to get to: 7n+1 - 3n+1
     
  2. jcsd
  3. Apr 23, 2013 #2
    Here is how I would go about it. Since we care about 7^n and 3^n mod 4, it might be helpful to write 7^n=4a+r and 3^n=4b+r where r is the remainder when 7^n (or 3^n) is divided by 4 (Why must r be the same for both expressions?)
    Write 7^(n+1)=7*7^n and 3^(n+1)=3*3^n and substitute.
     
    Last edited: Apr 23, 2013
  4. Apr 23, 2013 #3

    LCKurtz

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    Try not substituting that second term so you have$$
    7(4x+3^n) - 3\cdot 3^n$$and see if you can factor out a 4 from that.
     
  5. Apr 23, 2013 #4

    Dick

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    You probably meant to write 4|(7^n-3^n). Not '/'. The '|' means 'divides'.
     
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