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    in the actual exercises in the P6 Edexcel book, I seemed to do ok on these questions, but the exam ones seem so different... can anyone help me with methods?

    eg taking

    f(n) = (2n + 1)7^n - 1

    and trying to prove by induction that for all positive integers n, f(n) is divisible by 4.
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    (Original post by lesser weevil)
    in the actual exercises in the P6 Edexcel book, I seemed to do ok on these questions, but the exam ones seem so different... can anyone help me with methods?

    eg taking

    f(n) = (2n + 1)7^n - 1

    and trying to prove by induction that for all positive integers n, f(n) is divisible by 4.
    Heh, i've just done this question a few minutes ago.

    With questions like this it's often useful to make all the powers you same.

    You can rewrite 7^(k+1) as 7[7^(k)] so here f(k+1)-f(k) turns out to be of the form (4)(7^k)(...), which obviously shows if divisible by 4 for n=k then it is also divisible by 4 for n=k+1.

    Generally when proving divisiblility by induction there are two ways to start the 'if true for n=k, then true for n=k+1' part:
    (i) Preferably evaluate f(k+1)-f(k).
    (ii) If the above doesn't seem to give you a nice expression, start from f(k+1), rewrite powers in terms of those seem in f(k), then add and subtract so that while you aren't altering the expression, some of the terms are replaced by a multiple of f(k), but the terms remaining are clearly multiples of the appropriate integer.
 
 
 
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