  Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity
$\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$,
Erd\"os and Graham asked whether
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any
sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq
c>1$. We resolve the transcendence counterpart of their question: as a special
case of our main theorem, we have that
$\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when
$\frac{n_{k+1}}{n_k}\geq c>2$. The bound $c>2$ is best possible thanks to the
identity at the beginning. This paper provides a new way to apply the Subspace
Theorem to obtain transcendence results and extends previous non-trivial
results obtainable by only Mahler's method for special sequences of the form
$n_k=d^k+r$.
