$10 \cdot 66$ There is a sequence, its first term is 1, for all $n \geqslant 2$, the product of the first $n$ terms of this sequence is $n^{2}$. The sum of the third term and the fifth term of this sequence is
(A) $\frac{25}{9}$.
(B) $\frac{31}{15}$.
(C) $\frac{61}{16}$.
(D) $\frac{576}{225}$.
(E) 34 .
(30th American High School Mathematics Examination, 1979)
[Solution] Let $a_{n}$ denote the $n$-th term of the sequence. According to the problem and noting that
$$
a_{n}=\frac{a_{1} a_{2} \cdots a_{n}}{a_{1} a_{2} \cdots a_{n-1}}=\frac{n^{2}}{(n-1)^{2}}(n \geqslant 2),
$$

the first five terms of the sequence are $1,4, \frac{9}{4}, \frac{16}{9}, \frac{25}{16}$, so the required sum is
$$
\frac{9}{4}+\frac{25}{16}=\frac{61}{16} \text {. }
$$

Therefore, the correct choice is (C).