1976 AHSME Problem 21

What is the smallest positive odd integer n such that the product 2^{1/7}2^{3/7}\cdots2^{(2n+1)/7} is greater than 1000? (In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from 1 to 2n+1.)

\textbf{(A) }7\qquad \textbf{(B) }9\qquad \textbf{(C) }11\qquad \textbf{(D) }17\qquad \textbf{(E) }19

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.