Prove 1^3 + 2^3 + ... + n^3 = n^2(n + 1)^2/4 by induction

Step by step

The question

Question: prove by induction that 1³ + 2³ + … + n³ = n²(n + 1)² over 4.

Induction works like dominoes.

Push the first one: it is true for n = 1.

Then show each one knocks over the next: if it is true for k, it is true for k + 1.

Then they all fall.

1 Check n = 1

For n = 1 the left side is just the first number, 1.

The right side with n = 1 is also 1.

The first domino falls.

2 Assume it works for k

Now suppose it works for some whole number k: the sum up to k equals the formula with k.

3 What we must show

We must show it for k + 1: one more term, (k + 1)³, and every n on the right becomes k + 1.

Tidy the brackets.

Careful!

k + 1 goes into every n of the formula.

Adding 1 at the end is not the same.

4 Use the assumption

The left side starts with the sum up to k.

By our assumption, that is the formula with k.

We swap it in.

5 Make both sides the same

Same bottom: (k + 1)³ is 4 times it, over 4.

Open the brackets.

Like terms together.

Now the right side, the one we must get.

The square means the bracket times itself.

Open the brackets.

Like terms together.

The same as the left side!

So if it works for k, it works for k + 1.

It is true for n = 1, and each n passes it on to the next.

So it is true for every whole number n.

The proof is done.

6 Check

Let's check n = 4 by adding: 1 + 8 + 27 + 64 is 100.

And the formula with 4: also 100.

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: make both sides the same.

Last: check.