AHSME 1979 Problem 14

In a certain sequence of numbers, the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^2$. The sum of the third and the fifth numbers in the sequence is $\textbf{(A) }\frac{25}{9} \textbf{(B) }\frac{31}{15} \textbf{(C) }\frac{61}{16} \textbf{(D) }\frac{576}{225} \textbf{(E) }34$