South African Mathematics Olympiad

Points $A_1$, $A_2$, $A_3$ ... are constructed as follows: the length $O{A}_1$ is $4$, $\angle O{A}_1{A}_2=90^{\circ }$ and the length $A_1{A}_2=1$; then a right angle is constructed at $A_2$ to find $A_3$, and so on as shown in the diagram. The length of $O{A}_{21}$ is