Finding N formula (Backward Difference)

Question

I want to find the N formula of $▽^n f(x)=x^2+x$ Where $h=1$ So far, I reached to this point,

$▽ f(x)=f(x)-f(x-h)$

$ =x^2+x-[(x-h)^2+(x-h)]$

$= x^2+x-[x^2-2xh+h^2+x-h]$

$= x^2+x-x^2+2xh-h^2-x+h$

$= 2xh-h^2+h$

What to do next?

Answer 1

The binomial theorem appears in one of its many guises here.

$▽ ^{(n)} f(x) =\sum_{k=0}^n (-1)^k\binom{n}{k}f(x-kh) $.

This can (and should, as an exercise) be proved by induction.

Then get $f(x-kh)$ and substitute.