Given that f(x)= 7f(x-1) - 14f(x-2) + 8f(x-3) and that f(0)=0, f(½)=1, and f(1)=3, find f(x).
The equation f(x)= 7f(x-1)-14f(x-2)+8f(x-3) is similar to linear recurrences equations, so we can solve it the same way. Fiirst, solve x^3=7x^2-14x+8, which has roots 1, 2, and 4. This tells us that the solution is something like a.1^x+b.2^x+c.4^x. Now look at the three known values, and we find a=-1, b=0, and c=1, so the solution is
f(x)=4^x-1.
Solution
