Add up the first 100 odd numbers

Question

Add up the first 100 odd numbers: 1 + 3 + 5 + 7 + \cdots + 199 = \ ? Find the total without slogging through 100 additions. There is a surprisingly clean answer.

Accepted Answer

The sum is 10,000, exactly 100^2. Spot the pattern with small cases. Add the odd numbers a few at a time and watch the running totals: - 1 = 1 = 1^2 - 1 + 3 = 4 = 2^2 - 1 + 3 + 5 = 9 = 3^2 - 1 + 3 + 5 + 7 = 16 = 4^2 The sum of the first n odd numbers is always n^2, a perfect square. So the first 100 odd numbers sum to 100^2 = \mathbf{10{,}000}. Why it's a perfect square (a picture). Picture building a square out of dots, one L-shaped layer at a time. A 1	imes1 square is 1 dot. To grow it to 2	imes2 you add an L of 3 dots. To reach 3	imes3 you add an L of 5 dots. Each new layer wrapping aro…

Add up the first 100 odd numbers

Your answer

More Brainteasers questions