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Fp1 proof by mathematical induction that a expression is divisible by an interger... Watch

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    First you show that f(1) is divisible by the interger, then assume f(k) is divisible by the interger.

    In my book, for proving f(k+1) is divisible by the interger, they show this in all the examples:

    f(k+1) - f(k) = whatever those values are substituted in

    does it have to be f(k+1) - f(k) since it doesn't seem to be divisible by the interger for some questions I'm doing. Can it instead be e.g. f(k+1) + f(k), or would that no longer be proof?
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    (Original post by perrytheplatypus)
    First you show that f(1) is divisible by the interger, then assume f(k) is divisible by the interger.

    In my book, for proving f(k+1) is divisible by the interger, they show this in all the examples:

    f(k+1) - f(k) = whatever those values are substituted in

    does it have to be f(k+1) - f(k) since it doesn't seem to be divisible by the interger for some questions I'm doing. Can it instead be e.g. f(k+1) + f(k), or would that no longer be proof?

    Showing f(k+1)+f(k) is divisible by what ever (lets call it n) would work. Since f(k) is divisible by n, and you've show that f(k+1)+f(k) is divisble by n, then f(k+1) = "f(k+1)+f(k)" - f(k) would also be divisible by n; you would need to add that step to your working.

    HOWEVER, f(k+1)-f(k) will also be divisible by n, so can you give an example where you think it isn't.
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    (Original post by ghostwalker)
    Showing f(k+1)+f(k) is divisible by what ever (lets call it n) would work. Since f(k) is divisible by n, and you've show that f(k+1)+f(k) is divisble by n, then f(k+1) = "f(k+1)+f(k)" - f(k) would also be divisible by n; you would need to add that step to your working.

    HOWEVER, f(k+1)-f(k) will also be divisible by n, so can you give an example where you think it isn't.
    f(n)=7^n+4^n+1
    divisible by 6

    I get f(k+1) - f(k) = 6x7^k + 3x4^k

    but 3 isn't divisible by 6, so I'm stuck.
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    (Original post by perrytheplatypus)
    f(n)=7^n+4^n+1
    divisible by 6

    I get f(k+1) - f(k) = 6x7^k + 3x4^k

    but 3 isn't divisible by 6, so I'm stuck.
    But 4^k is divisible by 2, so 3 \times 4^k is divisible by...
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    (Original post by nuodai)
    But 4^k is divisible by 2, so 3 \times 4^k is divisible by...
    ...I feel like such an idiot, but thanks.
 
 
 
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