Prove 5^n - 1 is divisible by 4 by induction Step by Step

Step by step

The question

Question: prove by induction that 5ⁿ − 1 is divisible by 4, for every whole number n.

5ⁿ − 1 is divisible by 4

Like dominoes: show it for n = 1, then show that k passes it on to k + 1.

1 Check n = 1

For n = 1: 4, and that is 4 × 1.

Divisible by 4: the first domino falls.

5¹ − 1 = 4 = 4 × 1

2 Assume it works for k

Suppose it works for k: 5ᵏ − 1 is a multiple of 4.

So it is 4 times some whole number.

Call it m.

5^k − 1 = 4m

m is a whole number

Move the rest to the other side: 5ᵏ equals 4m + 1.

We will need this.

5^k = 4m + 1

3 What we must show

We must show the same for k + 1: every n becomes k + 1.

= 5^{k + 1} − 1 is divisible by 4

Careful!

One more power means one more times the number.

It is 5 times as big, not 1 more.

Careful: not 5^{k + 1} = 5^k + 1 ✗

4 Use the assumption

One more power: 5 times the power with k.

5^{k + 1} − 1 = 5 × 5^k − 1

5^{k + 1} = 5 × 5^k

Now swap in 4m + 1 for 5ᵏ.

= 5(4m + 1) − 1

Open the bracket.

= 20m + 5 − 1

Like terms together.

= 20m + 4

5 Take out 4

Every term has a 4 in it.

Take out 4: 4(5m + 1).

= 4(5m + 1)

5m + 1 is a whole number

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.

divisible by 4 for n = 1, and k → k + 1: for every n. ∎

6 Check

Let's check n = 3: it gives 124.

n = 3: 5³ − 1 = 124

124 is 4 times 31.

Divisible by 4.

Right!

124 = 4 × 31

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 4.

Last: check.

Answer