The sequence 1, 3, 6, 10, 15, etc find dy/dx...
dy/dx=2,3,4,5 this is not a constant so find d2y/dx2
d2y/dx2=1,1,1, this is a constant, specifically a constant acceleration, which is a property of a quadratic function of the form ax^2+bx+c and if we use three points of the sequence we can solve for all of the variables...
9a+3b+c=6
4a+2b+c=3
a+b+c=1 getting differences
5a+b=3
3a+b=2 and again
2a=1, so a=1/2, making 3a+b=2 become:
1.5+b=2, so b=1/2, making a+b+c=1 become
1/2+1/2+c=1 so c=0 so our function is:
a(n)=(n^2+n)/2
which can be expressed, if you factor the numerator as:
a(n)=n(n-1)/2
