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Math Help - Proof by induction

  1. #1
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    Proof by induction

    Hello guys could anyone help find a solution to this problem,
    Show that:

    3 x 5^(2n-1) + 2^(3n-2) is divisble by 17 (for every n on N*)


    This is what i did :
    verification for n = 1.
    then we assume it is divisible for n and show it is also divisble for n+1

    3 x 5^(2n+2-1) + 2^(3n+3-2)
    = 3 x 5^(2n+1) + 2^(3n+1)
    = 3 x 5^(2n-1) x 5^2 + 2^(3n-2) x 2^3
    i tried a lot of things but i couldnt find the solutions.
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  2. #2
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    Re: Proof by induction

    Let the statement be true for n = k
    That means 3 * 5^(2k-1) + 2^(3k-2) = 17 p for some p in N
    That gives 3 * 5^(2k-1) = 17 p - 2^(3k-2) ------- [ 1 ]
    Now for n = k+1 we have
    3 * 5^{2(k+1) -1} + 2^{3(k+1)-2} = 3 * 5^(2k+1) + 2^(3k+1) = 3 * 5^(2k-1+2) + 2^(3k-2+3)=3 * 25 *5^(2k-1) + 8*2^(3k-2)
    = 25* { 17 p - 2^(3k-2)} + 8*2^(3k-2) = 25* 17 p - 25* 2^(3k-2) + 8*2^(3k-2)= 25* 17 p - 17* 2^(3k-2)
    And you have the result
    Thanks from Orpheus
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  3. #3
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    Re: Proof by induction

    Quote Originally Posted by ibdutt View Post
    Let the statement be true for n = k
    That means 3 * 5^(2k-1) + 2^(3k-2) = 17 p for some p in N
    That gives 3 * 5^(2k-1) = 17 p - 2^(3k-2) ------- [ 1 ]
    Now for n = k+1 we have
    3 * 5^{2(k+1) -1} + 2^{3(k+1)-2} = 3 * 5^(2k+1) + 2^(3k+1) = 3 * 5^(2k-1+2) + 2^(3k-2+3)=3 * 25 *5^(2k-1) + 8*2^(3k-2)
    = 25* { 17 p - 2^(3k-2)} + 8*2^(3k-2) = 25* 17 p - 25* 2^(3k-2) + 8*2^(3k-2)= 25* 17 p - 17* 2^(3k-2)
    And you have the result
    Thanks.
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  4. #4
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    Re: Proof by induction

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