A Formula For Triangle Numbers

Let`s try and find a formula for the triangle numbers, assuming there is a quadratic formula an²+bn+c

(suitable for Higher level GCSE year 10 and 11, or `A` level)

the triangle numbers are 1, 3, 6, 10, 15, 21 etc...

if n=1, a+b+c=1 (1)

if n=2, 4a+2b+c=3 (2)

if n=3, 9a+3b+c=6 (3)

formula (2) - formula (1) leads to 3a+b=2

formula (3) - formula (2) leads to 5a+b=3

Solving as simultaneous equations 2a=1, so a=0.5 and b=0.5, from (1) c=0

so the formula for the n^th triangle number is 0.5n²+0.5n or 0.5n(n+1)