Linear recurrence problem

Question

Solve linear recurrence

f(1) = 12, f(2) = 16,

f(n) = 4f(n − 2).

Hint: the solution has integer coefficients, so if you get square roots and/or difficult fractions, it’s likely that you have made a mistake somewhere.

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So far i got:

f(n) = x²

f(n) = 4f(n - 2)
  x² = 4x^(n-2)
     = 4x^(2-2)
     = 4

Usually I get an quadratic equation or something with the multiplicity of 2, but I got a single integer, how do I solve this?

Answer 1

Outline

Assume $r^n$ is a solution, then the given condition reduces the equation to $r^n = 4r^{n-2}$ which means $r^2 = 4$, or $r = \pm 2$. This is where you are kind of stuck.

Now the general solution is given by $f(n) = a2^n + b(-2)^n$.

Given $f(1) = 12$ and $f(2) = 16$, you will get $a = 5$ and $b = -1$ so the most general solution is given by

$\color{blue}{f(n) = 5\times2^n - (-2)^n}$