$10 \cdot 47$ In a geometric sequence with positive terms, the difference between the fifth term and the fourth term is 576, and the difference between the second term and the first term is 9. The sum of the first five terms of this sequence is
(A) 1061.
(B) 1023.
(C) 1024.
(D) 768.
(E) None of these.
(25th American High School Mathematics Examination, 1974)
[Solution] The sum of the first five terms of a geometric sequence with the first term $a$ and common ratio $r$ is
$$
S_{5}=a+a r+a r^{2}+a r^{3}+a r^{4}=\frac{a\left(1-r^{5}\right)}{1-r} .
$$

Given $a r^{4}-a r^{3}=576$, and $a r-a=9$.
Dividing the first equation by the second equation gives $\frac{r^{4}-r^{3}}{r-1}=64$,
so $r^{3}=64$, which means $r=4$.
Since $a r-a=9$ and $r=4$, we have $a=3$, and $S_{5}=\frac{3\left(1-4^{5}\right)}{-3}=1023$. Therefore, the answer is $(B)$.