Sum of Squares of Consecutive Fibonacci Numbers
Theorem
- ${F_n}^2 + {F_{n + 1} }^2 = F_{2 n + 1}$
where $F_n$ denotes the $n$th Fibonacci number.
Proof
| \(\ds {F_n}^2 + {F_{n + 1} }^2\) | \(=\) | \(\ds F_{n + 1} F_{n - 1} - \paren {-1}^n + F_{n + 2} F_n - \paren {-1}^{n + 1}\) | Cassini's Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds F_n F_{n + 2} + F_{n - 1} F_{n + 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds F_{n + n + 1}\) | Honsberger's Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds F_{2 n + 1}\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $18$