Odd Numbers Sum to a Square (1+3+5+…=n²)
Add the odd numbers 1, 3, 5, 7, … and you always get a perfect square. Wrapping L-shaped gnomons around a square shows why.
Do not write an equation yet. Shake the figure first.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
Start with a 1×1 square (1 = 1²).
Return to the exact question
Show that the sum of the first n odd numbers 1+3+5+…+(2n−1) equals n², by growing a square.
Why this approach works
Instead of summing, grow a square layer by layer. Start with 1×1 (=1²). Wrap an L-shaped gnomon to reach 2×2 — that adds 3 cells. The next gnomon adds 5, then 7, …; growing k×k to (k+1)×(k+1) always adds 2k+1 cells, an odd number. So adding 1+3+5+… is the same as growing the square, and after n terms the n×n square is full: 1+3+5+…+(2n−1) = n².
Proof
Growing a k×k square to (k+1)×(k+1) adds a gnomon of 2k+1 cells (odd). Summing the odd numbers is thus growing the square, and after n terms it is n×n.
Try it yourself
Count each gnomon: 1, 3, 5, 7, 9 — all odd. Together they fill the 5×5 = 25 square.
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