1. The problem statement, all variables and given/known data Let P(n): 7|(34n+1-52n-1. Prove that P(n) is true for every natural number n. 2. Relevant equations *I know that proving by induction requires a proving P(1) true, and then proving P(k+1) true. *If a|b, then b=a*n, for some n∈ℤ 3. The attempt at a solution I have proved the "base case" for P(1). Long story short, for n=1, you wind up with 7|238 by definition of divisibility since 238=7(34). So P(1) is true. The next part is where it gets tricky - Assuming P(k) is true, that is, 7|(34k+1-52k-1, which is the inductive hypothesis. Then proving for P(k+1). What I have so far is... 7|7|(34(k+1)+1-52(k+1)-1 =34k+5-52k+1 =34 * 34k+1-51*52k =34 * 34k+1-(3*52k+2*52k) =34 * 34k+1-3*52k-2*52k =.............. Now I don't know what to do! I know I have to get it to the point where I can use the inductive hypothesis, which is what I'm trying to do, but I've hit a wall, or taken a wrong turn when split the 5 into a two and a three. Any ideas?