TMUA 2017 · Paper 2 · Question 15

The function $f$ is defined on the positive integers as follows:

$f (1) = 5$ , and for $n \geq 1$ :

{f (n + 1) = 3 f (n) + 1 f (n + 1) = \frac{1}{2} f (n) if f (n) is odd if f (n) is even

The function $g$ is defined on the positive integers as follows:

$g (1) = 3$ , and for $n \geq 1$ :

{g (n + 1) = g (n) + 5 g (n + 1) = \frac{1}{2} g (n) if g (n) is odd if g (n) is even

What is the value of $f (1000) - g (1000)$ ?