Step by step
The question
Question: prove by induction that 2³ⁿ − 1 is divisible by 7, 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: 7, and that is 7 × 1.
Divisible by 7: the first domino falls.
2 Assume it works for k
Suppose it works for k: 2³ᵏ − 1 is a multiple of 7.
So it is 7 times some whole number.
Call it m.
Move the rest to the other side: 2³ᵏ equals 7m + 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 8 times as big, not 1 more.
4 Use the assumption
One more power: 2³ times the power with k.
Now swap in 7m + 1 for 2³ᵏ.
Open the bracket.
Like terms together.
5 Take out 7
Every term has a 7 in it.
Take out 7: 7(8m + 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 511.
511 is 7 times 73.
Divisible by 7.
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 7.
Last: check.