The Fibonacci numbers are defined such that $f_n = f_{n-1}+f_{n-2}$. Prove that for all $n \ge 1$, $f_1^2 + f_2^2 + f_3^2 + \cdots + f_n^2 = f_n \cdot f_{n+1}$.
$P(n) = f_1^2 + f_2^2 + f_3^2 + \cdots + f_n^2 = f_n \cdot f_{n+1}$
Base case: $1^2 = 1 \cdot 1$
Inductive step - need to prove $P(n + 1)$:
- $f_1^2 + f_2^2 + f_3^2 + \cdots + f_n^2 + f_{n+1}^2 = f_{n+1} \cdot f_{n+2}$
- $f_n \cdot f_{n + 1} + f_{n+1}^2 = f_{n+1} \cdot f_{n+2}$ (by the inductive hypothesis)
- $f_{n+1}(f_n + f_{n+1}) = f_{n+1}(f_n + f_{n +1 })$
I'm wondering if I took the correct steps in this proof by induction to reach the conclusion. If there are any quicker methods to reach the conclusion, then any feedback is appreciated.