Use induction to prove that for all positive integers n, f(n)=2^(3n+1)+3x5^(2n+1) is divisible by 17.

Prove the basis to be true. Let n=1 and this gives f(1)=16+375=391 which is divisible by 17. Now assume that if we let n=k f(k) is divisible by 17. If we now let n=k+1 and prove f(k+1) is divisible by 17 we have proven the statement. Using f(k+1) won't give an answer, but if we subtract f(k) from f(k+1) we can rearrange the formula to get f(k+1)=8xf(k)+17x3x5^(2k+1). If the statement is true for n=k then we have shown it's true for n=k+1 and it is also true for n=1. Therefore it is true for all positive integers of n.