6. Compare the numbers $\frac{100}{101} \times \frac{102}{103} \times \ldots \times \frac{1020}{1021} \times \frac{1022}{1023}$ and $\frac{5}{16}$
Answer: $\frac{100}{101} \times \frac{102}{103} \times \ldots \times \frac{2020}{2021} \times \frac{2022}{2023}<\frac{5}{16}$

Solution: Notice that $\frac{n}{n+1}<\frac{n+1}{n+2}$. Therefore, $A=\frac{100}{101} \times \frac{102}{103} \times \ldots \times \frac{2020}{2021} \times \frac{2022}{2023}<\frac{101}{102} \times \frac{103}{104} \times \ldots \times$ $\frac{1021}{1022} \times \frac{1023}{1024}=B$. By multiplying and simplifying the fractions, we get $A \times B=\frac{100}{1024}=\frac{25}{256}=\left(\frac{5}{16}\right)^{2}$. On the other hand, since $A<B$, then $A^{2}<A \times B=\left(\frac{5}{16}\right)^{2}$.