Answer with step-by-step explanation:
We are given that the recurrence relation
for n=5,6,7,..
Initial condition
We have to show that Fibonacci numbers satisfies the recurrence relation.
The recurrence relation of Fibonacci numbers
,
Apply this
Substitute n=2
Hence, the Fibonacci numbers satisfied the given recurrence relation .
Now, we have to show that is divisible by 5 for n=1,2,3,..
Now replace n by 5n
Apply induction
Substitute n=1
It is true for n=1
Suppose it is true for n=k
is divisible 5
Let
Now, we shall prove that for n=k+1 is true
It is multiple of 5 .Therefore, it is divisible by 5.
It is true for n=k+1
Hence, the is divisible by 5 for n=1,2,3,..