the Fibbonacci numbers are F(0) = 0 , F(1) = 1, F(n) = F(n-1)+F(n-2) for all n greater than or equal to 2.
so, the sequence is 0,1,1,2,3,5,8,13,21,---
if we can show that F(n-1)+F(n-2) = the given right hand side then we are through.
for, consider 5F(n-4)+3F(n-5) = 2F(n- 4)+{ 3F(n-4) +3F(n-2)}
= 2F(n-4)+3F(n-3) { since F(k-1)+F(k-2) = F(k) }
= F(n-3)+{ 2F(n-3)+2F(n-4)}
= F(n-3)+2F(n-2)
= F(n-2)+{ F(n-2)+F(n-3)}
=F(n-1)+F(n-2)
= F(n)
so, the fibanocci number can be found with the help of the formula 5F(n-4)+3F(n-5) for all n greater than or equal to 6.