Step by step
The question
Question: prove by induction that 9ⁿ − 1 is divisible by 8, 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: 8, and that is 8 × 1.
Divisible by 8: the first domino falls.
2 Assume it works for k
Suppose it works for k: 9ᵏ − 1 is a multiple of 8.
So it is 8 times some whole number.
Call it m.
Move the rest to the other side: 9ᵏ equals 8m + 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 9 times as big, not 1 more.
4 Use the assumption
One more power: 9 times the power with k.
Now swap in 8m + 1 for 9ᵏ.
Open the bracket.
Like terms together.
5 Take out 8
Every term has an 8 in it.
Take out 8: 8(9m + 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 728.
728 is 8 times 91.
Divisible by 8.
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 8.
Last: check.