Prove 7^n - 1 is divisible by 6 by induction

Step by step

The question

Question: prove by induction that 7ⁿ − 1 is divisible by 6, for every whole number n.

Like dominoes: show it for n = 1, then show that k passes it on to k + 1.

1 Check n = 1

For n = 1: 6, and that is 6 × 1.

Divisible by 6: the first domino falls.

2 Assume it works for k

Suppose it works for k: 7ᵏ − 1 is a multiple of 6.

So it is 6 times some whole number.

Call it m.

Move the rest to the other side: 7ᵏ equals 6m + 1.

We will need this.

3 What we must show

We must show the same for k + 1: every n becomes k + 1.

Careful!

One more power means one more times the number.

It is 7 times as big, not 1 more.

4 Use the assumption

One more power: 7 times the power with k.

Now swap in 6m + 1 for 7ᵏ.

Open the bracket.

Like terms together.

5 Take out 6

Every term has a 6 in it.

Take out 6: 6(7m + 1).

So it works for k + 1 too.

True for n = 1, and passed on each time: true for every whole number n.

The proof is done.

6 Check

Let's check n = 3: it gives 342.

342 is 6 times 57.

Divisible by 6.

Right!

Let's look back.

First: check n = 1.

Then: assume it works for k.

Then: what we must show.

Then: use the assumption.

Then: take out 6.

Last: check.