Step by step
The question
Question: prove by induction that 8ⁿ − 3ⁿ is divisible by 5, 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: 5, and that is 5 × 1.
Divisible by 5: the first domino falls.
2 Assume it works for k
Suppose it works for k: 8ᵏ − 3ᵏ is a multiple of 5.
So it is 5 times some whole number.
Call it m.
Move the rest to the other side: 8ᵏ equals 3ᵏ + 5m.
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 8 times as big, not 1 more.
4 Use the assumption
One more power: 8 times the power with k.
Now swap in 3ᵏ + 5m for 8ᵏ.
Open the bracket.
Like terms together.
5 Take out 5
Every term has a 5 in it.
Take out 5: 5(3ᵏ + 8m).
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 485.
485 is 5 times 97.
Divisible by 5.
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 5.
Last: check.