The nth odd number

The idea

The nth odd number is 2n – 1, so the square n² is obtained by adding 2n – 1 to the previous square (n – 1)².

From 35² to 36²

Suppose you know that 35² = 1225, which is the sum of the first 35 odd numbers.

To get 36², which odd number do you add?

• 36
• 71
• 72

Count the L

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

2n − 1

The nth odd number is 2n − 1, so n² is the previous square (n − 1)² plus 2n − 1.

The L has (n − 1) + (n − 1) + 1 = 2n − 1 tiles. For n = 36: the 36th odd number is 2 × 36 − 1 = 71.

• Step | Working
• 35² | 1225
• 36th odd number | 2 × 36 − 1 = 71
• 36² | 1225 + 71 = 1296

Notes

The nth odd number is 2n − 1, so n² = (n − 1)² + (2n − 1).

Check yourself

What is the 20th odd number?

Answer: 39

2 × 20 − 1 = 39.

Which number do we add to 14² to get 15²?

49² = 2401. Use the nth odd number to find 50².

Answer: 2500

The 50th odd number is 99, and 2401 + 99 = 2500 = 50².

• 28. 28 = 2 × 14 is even. Add the 15th odd number, 2 × 15 − 1.
• 29 — correct. Yes! The 15th odd number is 2 × 15 − 1 = 29. (196 + 29 = 225.)
• 30. 30 = 2 × 15 is even. We add an odd number: 2n − 1.
• 31. 31 is the 16th odd number. For 15² we need the 15th: 2 × 15 − 1.