Prove 1^2 + 2^2 + ... + n^2 = n(n + 1)(2n + 1)/6 by induction Step by Step

Step by step

The question

Question: prove by induction that 1² + 2² + … + n² = n(n + 1)(2n + 1)/6.

1² + 2² + … + n² = n(n + 1)(2n + 1)6

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.

1 = 1(1 + 1)(2 × 1 + 1)6 = 1

✓ n = 1 works

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.

1² + 2² + … + k² = k(k + 1)(2k + 1)6

we assume this

3 What we must show

We must show it for k + 1: one more term, (k + 1)², and every n on the right becomes k + 1.

1² + 2² + … + k² + (k + 1)² = (k + 1)((k + 1) + 1)(2(k + 1) + 1)6

Tidy the brackets.

= (k + 1)(k + 2)(2k + 3)6

we must get this

Careful!

k + 1 goes into every n of the formula.

Adding 1 at the end is not the same.

Careful: not k(k + 1)(2k + 1)6 + 1 ✗

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.

1² + 2² + … + k² + (k + 1)² = k(k + 1)(2k + 1)6 + (k + 1)²

5 Make both sides the same

Same bottom: (k + 1)² is 6 times it, over 6.

= k(k + 1)(2k + 1) + 6(k + 1)²6

Open the brackets.

= 2k³ + k² + 2k² + k + 6k² + 6k + 6k + 66

Like terms together.

= 2k³ + 9k² + 13k + 66

Now the right side, the one we must get.

Open the brackets.

(k + 1)(k + 2)(2k + 3)6 = 2k³ + 3k² + 4k² + 6k + 2k² + 3k + 4k + 66

Like terms together.

= 2k³ + 9k² + 13k + 66

The same as the left side!

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

= the left side ✓

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.

true for n = 1, and k → k + 1: true for every n ≥ 1. ∎

6 Check

Let's check n = 4 by adding: 1 + 4 + 9 + 16 is 30.

n = 4: 1 + 4 + 9 + 16 = 30

And the formula with 4: also 30.

Right!

4(4 + 1)(2 × 4 + 1)6 = 30

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.

Answer