Holiday Contest 2025 - Guts Round - Set 6 Problem 1

Given that the following expression:

\begin{aligned} \frac{1}{5} + \frac{2}{20} + \frac{3}{85} + \frac{4}{260} + \frac{5}{629} + \frac{6}{1300} + \frac{7}{2405} \end{aligned}

can be expressed in the form of \frac{m}{n}, where m and n are positive relatively prime integers, compute m + n.