When I was 12 or 13 (about 3 years ago), I found this way to derive the formula, which is quite similar but not the same as your triangle method.

x

xx

xxx

xxxx

xxxxx

The number of Xs is the sum of 1, 2, 3, 4, and 5. Here’s how I derived the formula (Xs are the important elements of the triangle, Os are used to show what I don’t count, parentheses are used to show what I eliminate, and N is the number you want the sum up to).

xxxxx N

This is the number you want the sum of.

xoooo

xxooo

xxxoo N^2

xxxxo

xxxxx

Squaring N, you get the triangle and some extra elements (which need to be eliminated)

xoooo

xxooo

xxxoo N^2-N

xxxxo

(xxxxx)

There are no Os you need to eliminate in the bottom row, so you can remove it and add it back later.

x(oooo)

xx(ooo)

xxx(oo) (N^2-N)/2

xxxx(o)

(xxxxx)

Half of the elements above the bottom row are Os, so dividing by two gives you the number of Xs.

x(oooo)

xx(ooo)

xxx(oo) (N^2-N)/2+N

xxxx(o)

xxxxx

Add the original bottom row. You now have the triangle.

Time for my favorite part: changing how an equation looks. (N^2-N)/2+N is too sloppy.

(N^2-N)/2+N Original Equation

N^2/2 -N/2+N Distribute

N^2/2+N/2 Add -N/2 to N

(N^2+N)/2 Factor out 1/2

(N)(N+1)/2 Factor out N

That’s how I thought about it at the time, when I was bored one night. Yes, I developed theorems when I was 13. I still haven’t stopped. I like that way of looking at it, and when I showed that to my math class, most everyone understood what I was doing. Just goes to show you that there are no end to the ways that you can derive a formula.