# Thread: Prove by induction that our formula is divisible by 7.

1. ## Prove by induction that our formula is divisible by 7.

Hi guys,

the question is in two parts and I am stuck on the second part.

In the first part we have to prove by induction that $6^n - 1$ is divisible by 5. This was done by setting $6^n - 1 = 5A (A > 0)$ and setting $6^{n+1} - 1 = B$ and showing that B was modified only by factors of 5 in the process of adding the (n+1) th term. I mention this only in case the solution to the second part of the question relies on the first part, which it doesn't seem to.

The second part says Prove that $8^n - 7n + 6$ is divisible by 7 for all integral n.

I tried the same method as for part 1 but I cannot get it to work.

I have $8^k - 7k + 6 = 7A$ for n = k

I have $8^{k+1} - 7(k+1) + 6 = B$ for n = k+1

but I cannot reduce the k+1 term to an equation in the form of the k term as I could in part 1. Can anyone do better?

2. ## Re: Prove by induction that our formula is divisible by 7.

$8^{k+1}-7(k+1)+6 = 8\cdot 8^k -7k-7 + 6 = (8^k-7k+6)+7\cdot 8^k-7 = 7A+7(8^k-1)$

3. ## Re: Prove by induction that our formula is divisible by 7.

It's so easy when you see the answer! I posted the question and thought "that will take them a few hours" and it is done in minutes! many thanks as usual.