Prove the following statement by mathematical induction.
$7^n-2^n$ is divisible by 5 for each integer $n \ge 1$
My attempt: Let the given statement be p(n).
(1) $7-2=5$ is divisible by 5 so p(1) is true.
(2) Suppose as an inductive hypothesis, $7^k-2^k$ is divisible by 5 for each integer $k \ge 1$. That is, p(k) is true for all integer $k \ge 1$
Then we must show that p(k+1) is true:
$7^{k+1}-2^{k+1}$=$7^{k+1}-(7-5)^{k+1}$
I'm stuck on this step. I think I have to separate a multiple of 5 from the above equation, but I can't.
Best Answer
Hint:
$$ 7^{k+1}-2^{k+1} = (2+5)7^k - 2\cdot2^k.$$
Or alternatively:
$$ 7^{k+1} - 2^{k+1} = 7\cdot 7^k - (7-5)2^k.$$
Or alternatively yet:
$$ 7^{k+1} - 2^{k+1} = 7\cdot 7^k - 2\cdot 2^k + 7\cdot 2^k - 7\cdot 2^k.$$
There are some more variations of this trick.