  We consider random Fibonacci sequences given by $x_{n+1}=\pm \beta
x_{n}+x_{n-1}$. Viswanath (\cite{viswanath}), following Furstenberg
(\cite{furst}) showed that when $\beta = 1$, $\lim_{n\to
\infty}|x_{n}|^{1/n}=1.13...$, but his proof involves the use of floating point
computer calculations. We give a completely elementary proof that $1.25577 \ge
(E(|x_{n}|))^{1/n} \ge 1.12095$ where $E(|x_{n}|)$ is the expected value for
the absolute value of the $n$th term in a random Fibonacci sequence. We compute
this expected value using recurrence relations which bound the sum of all
possible $n$th terms for such sequences. In addition, we give upper an lower
