Prove, using mathematical induction, that the given property of the Fibonacci numbers is true [F(n+1)]2 =[F(n)]2+F(n−1)∗F(n+2) foran≥2 Where Fibonacci numbers are defined by the recurrence relation F(1) = 1 F(2) = 1 F(n) = F(n − 1) + F(n − 2)

We have proven that the property of the Fibonacci sequence holds by Mathematical Induction.

Explanation:

For n≥2. We prove that it holds for n=2, Assume that it holds for n=k and prove that it holds for n=k+1

F(1)=1

F(2) = 1

F(3) = F(3 − 1) + F(3 − 2)= F(2) + F(1)=1+1=2

F(4) = F(4 − 1) + F(4 − 2)= F(3) + F(2)=2+1=3

In

We prove it is true for n=2.

When n=2

We assume it is true for n=k and prove that it holds for n=k+1.

When n=k+1 in

Substituting n=k+1 in the LHS: and applying: F(k+2)=F(k+1)+F(k)

Substituting n=k+1 in the RHS :

RHS=

Since the LHS=RHS

Therefore, the property is true.

FOR REFERENCE

When n=k in

Substituting n=k in the LHS: and applying: F(k+1)=F(k)+F(k-1)

LHS=RHS

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