(%i1) f(x) := a*x*x + b*x + c;
(%o2)                       f(x) := a x x + b x + c
(%i2) solve([ f(0) = 1, f(1) = 2, f(2) = 4]);
                                  1      1
(%o2)                       [[b = -, a = -, c = 1]]
                                  2      2

(%i3) g(x) := ( x + 1) * x / 2 + 1;
                                     (x + 1) x
(%o3)                        g(x) := --------- + 1
                                         2
(%i4) g(3);
(%o4)                                7
(%i5) g(4);
(%o5)                               11
(%i6) g(5);
(%o6)                               16

(%i7) expand(g(x) - g(x-1));
(%o7)                                  x