The sum of the first natural number and 0 is 1, the sum of the first two natural numbers and 0 is 3, the sum of the first three natural numbers and 0 is 6, and the sum of the first four natural numbers and 0 is 10. which explicit formula generates the sum of the first n natural numbers and 0?

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