How do I solve $f(x) - f(x - 1) = x$? Is it possible to solve the functional equation $f(x) - f(x - 1) = x$ by algebraic means?
First note that we can add a constant to $f$ and it will still be a solution, so we might as well define $f(0)=0$  Then we can find $f(1)=1, f(2)=3, f(3)=6$ and so on.  We can take finite differences and find that a quadratic polynomial fits all of these, then find that $\frac 12x(x+1)$ satisfies our equation.  The full solution is then $f(x)=\frac 12x(x+1)+f(0)$
Yes, just take any $f(0)$, and $f(n) = f(0)$ plus some summation.