Sum of Consecutive Odd Index Fibonacci Numbers

Theorem

$\ds F_{2 k - 1} + F_{2 k + 1}$

$=$

$\ds \phi^{2 k} + \phi^{-2 k}$

where:

$F_k$ denotes the $k$th Fibonacci number

$\phi$ denotes the golden mean.

$\ds F_{2 k - 1} + F_{2 k + 1}$

$=$

$\ds \paren {\phi^{2 k} - F_{2 k} \phi } + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi }$

$\ds $

$=$

$\ds \phi^{2 k} + F_{-2 k} \phi + \paren {\phi^{2 k + 2} - F_{2 k + 2} \phi }$

$\ds $

$=$

$\ds \phi^{2 k} + F_{-2 k} \phi + \paren {\paren {F_{2 k + 2} \phi + F_{2 k + 1} } - F_{2 k + 2} \phi }$

$\ds $

$=$

$\ds \phi^{2 k} + \paren {F_{-2 k} \phi + F_{-2 k - 1} }$

$\ds $

$=$

$\ds \phi^{2 k} + \phi^{-2 k}$

$\blacksquare$