Proving Induction Problem: 5|(7^k-2^k) for all k∈R

[skiing4free]

Homework Statement

Prove the following statement by induction

5|(7^k-2^k) for all k$$\in$$R


2. The attempt at a solution
Started by proving P(k)=5|(7^k-2^k) where k=0
which gives 5|0=7^0-2^0 =>0

Then when P(k+1)=5|(7^k-2^k) it gets more complicated and I get stuck.
Proving this induction: 7^(k+1)-2^(k+1)=7.7^k-2.2^k
But this is as far as a get and I cannot seem to get the nest step.

The answers show that the next steps are:
=> 7.7^k-7.2^k+7.2^k-2.2^k
=> 7(7^k-2^k)+(7-2)2^k
=> 5(7x+2^k)

Which apparently is the proof.

I'm confused as to how they get from the 1st step to the second and also where the variable x comes from in the final step.

Any help on this question would be greatly appreciated.

 

[Dick]

The induction assumption is that 7^k-2^k is divisible by 5. So it's equal to 5*x, where x is some integer. That means 7^k-2^k=5x. That's the x they are talking about.