Smallest integer in radical product sequence

The expression can be simplified as: $10^{\frac{1 + 2 + \dots + n}{7}} = 10^{\frac{n(n+1)}{14}} > 999$

Since $10^3 = 1000 > 999$, we need: $\frac{n(n+1)}{14} \ge 3 \implies n(n+1) \ge 42$

For $n = 5$: $5 \times 6 = 30 < 42$. For $n = 6$: $6 \times 7 = 42 \ge 42$.

Hence, the smallest value of $n$ is 6.