Theorem index16 results

Sum of Fibonacci Squares

Theorem 8. If $f_1,f_2,\ldots$ are the Fibonacci numbers, then for every $n\in\mathbb N$ , $f_1^2+f_2^2+\cdots+f_n^2=f_nf_{n+1}$ .

8.1ProofMathematical induction

MethodMathematical induction

1
Base case. For $n=1$ , $f_1^2=1=f_1f_2$ .
2
Inductive hypothesis. Assume the identity holds at $n=k$ .
3
Inductive step. Then $f_1^2+\cdots+f_k^2+f_{k+1}^2=f_kf_{k+1}+f_{k+1}^2=f_{k+1}(f_k+f_{k+1})=f_{k+1}f_{k+2}$ . Thus the identity holds for $k+1$ , and induction completes the proof.