IMAS 2022–2023 Junior Division — First Round Question 16

In the Fibonacci sequence $\{F_k\}$, $F_1=F_2=1$ and $F_{k+1}=F_k+F_{k-1}$ for $k\ge2$. What is the value of $\frac1{F_1F_3}+\frac1{F_2F_4}+\frac1{F_3F_5}+\frac1{F_4F_6}+\frac1{F_5F_7}+\frac1{F_6F_8}+\frac1{F_7F_9}+\frac1{F_8F_{10}}$?