Fibonacci Number of Index 2n as Sum of Squares of Fibonacci Numbers

Theorem

Let $F_n$ denote the $n$th Fibonacci number.

Then:

$F_{2 n} = {F_{n + 1} }^2 - {F_{n - 1} }^2$

Proof

From Honsberger's Identity:

$\forall m, n \in \Z_{>0}: F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

Setting $m = n$:

\(\ds F_{2 n}\)

\(=\)

\(\ds F_{n - 1} F_n + F_n F_{n + 1}\)

\(\ds \)

\(=\)

\(\ds F_n \paren {F_{n + 1} + F_{n - 1} }\)

\(\ds \)

\(=\)

\(\ds \paren {F_{n + 1} - F_{n - 1} } \paren {F_{n + 1} + F_{n - 1} }\)

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds {F_{n + 1} }^2 - {F_{n - 1} }^2\)

Difference of Two Squares

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $5$