# Sum of Reciprocals of Sequence of Pairs of Even Index Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared

## Theorem

 $\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }$ $=$ $\ds \dfrac 1 {1 \times 3} + \dfrac 1 {3 \times 8} + \dfrac 1 {8 \times 21} + \dfrac 1 {21 \times 55} + \cdots$ $\ds$ $=$ $\ds \phi^{-2}$

where:

$F_k$ denotes the $k$th Fibonacci number
$\phi$ denotes the golden mean.

## Proof

 $\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} }$ $=$ $\ds \sum_{k \mathop \ge 1} \dfrac 1 {F_{2 k} F_{2 k + 2} } \paren {\dfrac {F_{2 k + 2} - F_{2 k} } {F_{2 k + 2} - F_{2 k} } }$ multiplying by $1$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {F_{2 k + 2} - F_{2 k} } }$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {\paren {F_{2 k + 1} + F_{2 k} } - F_{2 k} } }$ Definition of Fibonacci Number $\ds$ $=$ $\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} } - \dfrac 1 {F_{2 k + 2} } } \paren {\dfrac 1 {F_{2 k + 1} } }$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 1} \paren {\dfrac 1 {F_{2 k} F_{2 k + 1} } - \dfrac 1 {F_{2 k + 1} F_{2 k + 2} } }$ $\ds$ $=$ $\ds \sum_{k \mathop \ge 2} \paren {-1}^k \dfrac 1 {F_k F_{k + 1} }$ $\ds$ $=$ $\ds \phi^{-2}$ Sum of Alternating Sign Reciprocals of Sequence of Pairs of Consecutive Fibonacci Numbers is Reciprocal of Golden Mean Squared

$\blacksquare$

## Historical Note

This result is attributed to Pincus Schub.