Triangular Numbers: 1+2+…+n = n(n+1)/2
A fast way to add 1+2+…+n? Stack it as a staircase, mate an upside-down copy, and you get a rectangle — the answer is half.
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.
Stack 1+2+3+4+5 as a staircase (triangular number T₅).
Return to the exact question
Find 1 + 2 + 3 + … + n (a triangular number) by mating staircases into a rectangle.
Why this approach works
Stack 1+2+…+n as a staircase. Make an identical copy, rotate it 180°, and mate it on top: the two jagged staircases lock into a perfect n by (n+1) rectangle. Two staircases fill n(n+1), so one is half — 1+2+…+n = n(n+1)/2. The geometric form of young Gauss’s pairing trick (1+100, 2+99, …).
Proof
A staircase and its 180°-rotated copy mate into an n×(n+1) rectangle. Two staircases make one rectangle, so 1+2+…+n = n(n+1)/2.
Try it yourself
For n=100 the rectangle is 100×101, half of which is 5050 — exactly Gauss’s mental answer.
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