Step by step
The question
Question: prove by induction that 10ⁿ − 1 is divisible by 9, 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: 9, and that is 9 × 1.
Divisible by 9: the first domino falls.
2 Assume it works for k
Suppose it works for k: 10ᵏ − 1 is a multiple of 9.
So it is 9 times some whole number.
Call it m.
Move the rest to the other side: 10ᵏ equals 9m + 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 10 times as big, not 1 more.
4 Use the assumption
One more power: 10 times the power with k.
Now swap in 9m + 1 for 10ᵏ.
Open the bracket.
Like terms together.
5 Take out 9
Every term has a 9 in it.
Take out 9: 9(10m + 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 999.
999 is 9 times 111.
Divisible by 9.
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 9.
Last: check.