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    Hi I had to prove by induction that 12^n +2 x 5^(n-1) is divisible by 7

    I have got to [12^(k+1) + 2 x 5^k] -5[12^k +2 x 5^(k-1)]

    Stupid thing is i dunno how to bloody multiply out -5[12^k +2 x 5^(k-1)]. Any help would be much appreciated! Step by step would be wonderful haha


    EDIT: do I multiply the -5 to the k-1 to make K?!
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    Instead of trying to expand your f(k) to be of the same index as f(k+1), consider doing it the other way around.
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    (Original post by bonana567)
    Hi I had to prove by induction that 12^n +2 x 5^(n-1) is divisible by 7

    I have got to [12^(k+1) + 2 x 5^k] -5[12^k +2 x 5^(k-1)]

    Stupid thing is i dunno how to bloody multiply out -5[12^k +2 x 5^(k-1)]. Any help would be much appreciated! Step by step would be wonderful haha


    EDIT: do I multiply the -5 to the k-1 to make K?!
     [12^{k+1} + 2\times 5^k] - 5[12^k +2\times5 ^{k-1}]

    well, it should have already been shown for
     [12^n +2\times5 ^{n-1}]
    so the right side divides.

    I think?

    _Kar.
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    f(k) = 12^k + 2x5^(k-1)
    f(k+1) = 12^(k+1) + 2x5^(k)

    f(k+1) - 5f(k)

    [12^(k+1) + 2x5^(k)] - 5[12^k + 2x5^(k-1)]
    = 12 x 12^k + 2x5^k - 5x12^k -2x5^k
    = 7x12^k (divisible by 7)

    Since by assumption 7 | f(k) is true
    then 7 | f(k+1) is also true

    therefore by PMI, 7 divides 12^k + 2x5^(k-1) etc etc.

    (sorry I can't write Latex, hope this helps
 
 
 
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