Question 16
Let
$$A_{m,n}=\frac1{\log_2 2025\,\log_3 2\,\log_4 3\cdots\log_{99}98\,\log_{2025^{mn}}99}$$
and let $S_k=A_{2025,2}+A_{2025,3}+\cdots+A_{2025,k}$. If $T=\sum_{k=2}^{\infty}1/S_k$, find $\lceil1/T\rceil$.
$$A_{m,n}=\frac1{\log_2 2025\,\log_3 2\,\log_4 3\cdots\log_{99}98\,\log_{2025^{mn}}99}$$
and let $S_k=A_{2025,2}+A_{2025,3}+\cdots+A_{2025,k}$. If $T=\sum_{k=2}^{\infty}1/S_k$, find $\lceil1/T\rceil$.