7. For a certain game activity,

the rewards are divided into first, second, and third prizes (all participants in the game activity will receive a prize), and the corresponding winning probabilities form a geometric sequence with the first term $a$ and a common ratio of 2. The corresponding prize money forms an arithmetic sequence with the first term of 700 yuan and a common difference of -140. Then the expected prize money for participating in this game activity is $\qquad$ yuan.
7.500 .

From the problem, we know the probabilities of winning the first, second, and third prizes are
$$
P_{1}=a, P_{2}=2 a, P_{3}=4 a .
$$

From $P_{1}+P_{2}+P_{3}=1$, we get $a=\frac{1}{7}$.
Thus, $P_{1}=\frac{1}{7}, P_{2}=\frac{2}{7}, P_{3}=\frac{4}{7}$.
The prizes for winning the first, second, and third prizes are
$$
\begin{array}{l}
\quad \xi_{1}=700, \xi_{2}=560, \xi_{3}=420 . \\
\quad \text { Therefore, } E \xi=700 \times \frac{1}{7}+560 \times \frac{2}{7}+420 \times \frac{4}{7} \\
=500 \text { (yuan). }
\end{array}
$$