Adding the nth odd number

The idea

The sum of the first n odd numbers is n² and the nth odd number is 2n – 1, so (n – 1)² + (2n – 1) = n².

Odd numbers make squares

Add the first few odd numbers: 1 + 3 + 5 = 9, which is 3².

Now add the first five odd numbers: 1 + 3 + 5 + 7 + 9.

What do you think the total is?

• 20
• 25
• 30

Add an L

Move n. The square (n − 1)² gets an L-shaped layer: two arms and a corner. Count the tiles in the L.

(n − 1)² + (2n − 1) = n²

The sum of the first n odd numbers is n², and the nth odd number is 2n − 1. So adding the nth odd number to (n − 1)² gives the next square: (n − 1)² + (2n − 1) = n².

For n = 6: the 6th odd number is 2 × 6 − 1 = 11, and 5² + 11 = 25 + 11 = 36 = 6².

You can also see it by expanding: (n − 1)² = n² − 2n + 1. Adding 2n − 1 leaves exactly n².

Notes

The sum of the first n odd numbers is n², and the nth odd number is 2n − 1. So (n − 1)² + (2n − 1) = n².

Check yourself

What is the 13th odd number?

Answer: 25

2 × 13 − 1 = 25.

Use the rule to find 12². You know 11² = 121. Add the 12th odd number.

Answer: 144

The 12th odd number is 2 × 12 − 1 = 23, and 121 + 23 = 144 = 12².

Which sum of odd numbers is equal to 7²?

The 9th odd number is 17. What is 8² + 17?

Answer: 81

8² + 17 = 64 + 17 = 81 = 9².

• 1 + 3 + 5 + 7 + 9 + 11. That has 6 odd numbers, so it is 6² = 36.
• 1 + 3 + 5 + 7 + 9 + 11 + 13 — correct. Yes! Seven odd numbers add to 7² = 49.
• 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15. That has 8 odd numbers, so it is 8² = 64.