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.