Why the First N Odd Numbers Add Up to N²
The first N odd numbers always sum to N².
1 + 3 + 5 + 7 + 9 = 25
Why do the first N odd numbers add up to N squared?
The first N odd numbers always add up to N². For the first five, 1 + 3 + 5 + 7 + 9 = 25, which is 5². Start with one dot, a 1 by 1 square. Add 3 dots in an L shape around one corner, and it becomes a 2 by 2 square. Add 5 more in the next L, and it becomes 3 by 3. Each L covers one new row and one new column that share a corner, so the Nth L holds N + N − 1 dots, which is the Nth odd number. After N of them the square is N by N. To find how many odd numbers you are adding, add 1 to the last one and halve: 19 + 1 = 20, and half of 20 is 10, so the odd numbers up to 19 add to 10², or 100.
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A 4 by 4 square sits inside a 5 by 5, and the L-shaped rim around it holds 9.
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Every odd number is one of those rims: 1, then 3, then 5, then 7, then 9.
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Five rims complete a 5 by 5 square, so .
Your turn
Three to try in your head — tap what you get.
1 + 3 + 5 + ... + 17
1 + 3 + 5 + ... + 19
1 + 3 + 5 + ... + 11
All 36 shortcuts: the Magic Tricks library.