Add the odd numbers one by one. The running total is always a perfect square. Try it: 1 = 1, 1+3 = 4, 1+3+5 = 9, 1+3+5+7 = 16. No coincidence.
Try this
6
1 + 3 + 5 + … + (2n − 1) = n² = 1 + 3 + 5 + 7 + 9 + 11 = 36
Your turn
What is 1 + 3 + 5 + 7 + 9 + 11 + 13?
Picture an n × n square of dots: peel it as nested L-shapes of 1, 3, 5, 7… dots. Each L is the next odd number — so the odds stack up to a perfect square.
Recap
- Running totals of odd numbers are exactly the squares: 1, 4, 9, 16, 25…
- 1 + 3 + … + (2n − 1) = n².
- Number of odd terms added = the square root of the total.
- Nicomachus knew this around 100 AD — and it's provable with dots.
Sum of first n odd numbers
Nicomachus knew this around 100 AD — and you can prove it by drawing dots.
Visualise it: an n × n square of dots can be split into L-shapes of 1, 3, 5, 7… dots each. Each L is one more odd number.