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# proving by induction P6 watch

1. 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.
2. (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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