Problem 21 (1976 AHSME)

What is the smallest positive odd integer $n$ such that the product $2^{1/7} 2^{3/7} \cdots 2^{(2n+1)/7}$ 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 $2n+1$.)