Step by step
The question
Question: prove by induction that n⁵ − n 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: 0, and that is 5 × 0.
Divisible by 5: the first domino falls.
2 Assume it works for k
Suppose it works for k: k⁵ − k is a multiple of 5.
So it is 5 times some whole number.
Call it m.
3 What we must show
We must show the same for k + 1: every n becomes k + 1.
Careful!
A bracket to a power is the bracket times itself.
Every part times every part.
4 Use the assumption
Put k + 1 in.
Open the brackets.
Like terms together.
Look for the old one inside: k⁵ − k.
The rest is left over.
By our assumption, the old one is 5m.
5 Take out 5
Every term has a 5 in it.
Take out 5: 5(k⁴ + 2k³ + 2k² + k + m).
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 240.
240 is 5 times 48.
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.