Show that the Fibonacci numbers satisfy the recurrence relation $f_{n} = 5f_{n−4} + 3f_{n−5} \:\text{for}\: n = 5, 6, 7,\dots,$ together with the initial conditions $f_{0} = 0, f_{1} = 1, f_{2} = 1, f_{3} = 2, \:\text{and}\: f4 = 3.$ Use this recurrence relation to show that $f_{5n}$ is divisible by $5$, for $n = 1, 2, 3,\dots .$
