Catalan's Identity/Proof 1

Theorem

${F_n}^2 - F_{n - r} F_{n + r} = \left({-1}\right)^{n - r} {F_r}^2$

Proof 1

From the definition of Fibonacci numbers:

$F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

By Honsberger's Identity:

$F_n = F_{n - r} F_{r - 1} + F_{n - r + 1} F_r$

Also:

\(\ds F_{n + r}\)

\(=\)

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

Honsberger's Identity

\(\ds \)

\(=\)

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

Honsberger's Identity

\(\ds \)

\(=\)

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

Definition of Fibonacci Number

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Therefore:

\(\ds \)

\(\)

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

\(\ds \)

\(=\)

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

from above

\(\ds \)

\(=\)

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

Expansion and Simplification

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Definition of Fibonacci Number

\(\ds \)

\(=\)

\(\ds {F_r}^2 \paren {-1}^{n - r}\)

Cassini's Identity

\(\ds \)

\(=\)

\(\ds \paren {-1}^{n - r} {F_r}^2\)

$\blacksquare$

Source of Name

This entry was named for Eugène Charles Catalan.