Chapter 8.1, Problem 28E

Show that the Fibonacci numbers satisfy the recurrence relation $f_n=5f_{n-4}+3f_{n+5}$ for $n=5,6,7,\mathrm{...}$ , together with the initial conditions $f_0=0,f_1=0,f_2=0,f_3=2$ , and $f_4=3$ . Use this recurrence relation to show that $f_{5n}$ is divisible by 5, for $n=1,2,3,\mathrm{...}$