2024 AMC 10B Problem 23

The Fibonacci numbers are defined by F_1 = 1, F_2 = 1, and F_n = F_{n-1} + F_{n-2} for n \ge 3. What is

\[\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}?\]

\textbf{(A)} 318 \qquad \textbf{(B)} 319 \qquad \textbf{(C)} 320 \qquad \textbf{(D)} 321 \qquad \textbf{(E)} 322

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.