here's a geometric "proof"

form a triangle (this is for n=4, notice 4 rows)

Code:Code:.
. .
. . .
. . . ..
. .
. . .
. . . .clearly there are 10 dots which is 4(4+1)/2.

Numbers which form these triangles are called triangular numbers.

take two of these triangles

Code:Code:. . . . . . . . . .
. . . . . -> . . . . .
. . . . . . . . . .
. . . . . . . . . .. . . . . . . . . .
. . . . . -> . . . . .
. . . . . . . . . .
. . . . . . . . . .
these form a rectangle when squeished together

but you can see that twice the sum of the triangle is [4 rows]*[(4+1) columns]

so 2(1+2+3+4) = 4(4+1), and therefore 1+2+3+4 = 4(4+1)/2

this is basically gaus's proof but using a picture

^ Brilliant. Haven't seen visual proof like that before.

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