A sequence of positive real numbers $\{a_1, a_2, a_3, \dots\}$ has the property that for $i \ge 2,$ each $a_i$ is equal to the sum of all the previous terms.  If $a_{19} = 99,$ then what is $a_{20}$?
For $n \ge 2,$
\[a_n = a_{n - 1} + a_{n - 2} + \dots + a_2 + a_1.\]Then
\begin{align*}
a_{n + 1} &= a_n + a_{n - 1} + a_{n - 2} + \dots + a_2 + a_1 \\
&= a_n + (a_{n - 1} + a_{n - 2} + \dots + a_2 + a_1) \\
&= 2a_n.
\end{align*}Hence, each term (starting with $a_2$) is double the last term, which means $a_{20} = 2 \cdot 99 = \boxed{198}.$