Sum of Odd Numbers From 1 to 99 in 5 Seconds (Formula: n²)

Step by step

1. 1 + 3 + 5 + … + 99 in 5 seconds?
   One plus three plus five, all the way to ninety-nine. Can you add them in five seconds?
2. 2500 · here's how to see it
   It's twenty-five hundred. Here's how to see it in a flash.
3. Way 1: odd numbers as tiles · start with 1
   Way one: turn each odd number into square tiles. Start with one tile.
4. + 3 in an L → 2 × 2
   Add three tiles around it, in an L shape. Now it's a two-by-two square.
5. + 5 → 3 × 3 · + 7 → 4 × 4
   Add five more in the next L: three by three. Then seven: four by four.
6. The L: n on top + n on the side + 1 corner
   Why does each L fit? Around an n-by-n square, the L needs n on top, n on the side, and one corner.
7. 2n + 1 = the next odd number → the next square
   That's two n plus one tiles: exactly the next odd number. And it makes the next square.
8. First n odd numbers → n × n
   So the first n odd numbers always add up to an n-by-n square.
9. 2 × 50 − 1 = 99 → the 50th odd number
   How many odd numbers go up to ninety-nine? Two times fifty, minus one, is ninety-nine. So it's the fiftieth odd number.
10. 50 × 50 = 2500
    Fifty odd numbers make a fifty-by-fifty square: twenty-five hundred!
11. The L = a "gnomon" (carpenter's square) · Pythagoreans' pebbles → squares
    That L shape has a Greek name: the gnomon, like a carpenter's square. The story goes that Pythagoras's followers laid out pebbles this way, and always got squares.
12. Galileo (1638): a rolling ball goes 1, 3, 5, 7 in equal times
    Two thousand years later, Galileo saw the same numbers in a rolling ball. Time it in equal steps, and the ball rolls one, three, five, then seven units.
13. Way 2: pair the first with the last
    Here's a second way to see it: the pairing trick. Pair the first number with the last.
14. 1 + 99 = 100 · 3 + 97 = 100 · every pair = 100
    One plus ninety-nine is a hundred. Three plus ninety-seven is a hundred too. Every pair makes a hundred.
15. 25 pairs × 100 = 2500
    Fifty numbers make twenty-five pairs. Twenty-five times a hundred is twenty-five hundred. Same answer!
16. 1 + 3 + 5 + … (n odd numbers) = n²
    So: add up the first n odd numbers, and you always get n squared.
17. On a test: 99 = 2 × 50 − 1 → 50² = 2500
    On a test, write it like this. Ninety-nine is the fiftieth odd number. The first n odd numbers add up to n squared. So the sum is fifty squared: twenty-five hundred.