Not-So-Dumb Fibonacci Sum #1

Number Theory Level 3

Find the value of $\sum_{n=1}^{\infty} \dfrac{F_{n+1}}{F_{n}F_{n+2}}$

where $F_n$ is the $n$ th Fibonacci number, defined by the recurrence relation $F_n=F_{n-1}+F_{n-2}$ with $F_1=F_2=1$ .

The answer is 2.

1 solution

Nick Diaco
Oct 13, 2015

$\sum_{n=1}^{\infty} \frac{F_{n+1}}{F_nF_{n+2}} = \sum_{n=1}^{\infty} \frac{F_{n+2}-F_n}{F_nF_{n+2}} = \sum_{n=1}^{\infty} \bigg( \frac{1}{F_n} - \frac{1}{F_{n+2}} \bigg) = \frac{1}{F_1} + \frac{1}{F_2} = 2$

Not-So-Dumb Fibonacci Sum #1

The answer is 2.

1 solution

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