AMC 10B/12B 2024 (Problem 23/18)

The Fibonacci numbers are defined by

F_1=1

F_2=1

, and

F_n=F_{n-1}+F_{n-2}

for

n\ge3

. What is

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

\text{(A)}\;318

\text{(B)}\;319

\text{(C)}\;320

\text{(D)}\;321

\text{(E)}\;322