Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 5A / b / iii / Solution

Let $d_n=x_{n+1}-x_n$ be the direction vector of the segment $X_n{X}_{n+1}$. Consecutive vectors $x_n,x_{n+1}$ are perpendicular and $|x_{n+1}|=\lambda |x_n|$. Using $x_{n+2}=-{\lambda }^2{x}_n$,Thus adjacent segments are perpendicular. AlsosoIf $0<\lambda <1$, the norm formula in part (i) gives $x_n\to 0$. If $\lambda =1$, then $x_{n+2}=-x_n$: the points $X_1,X_2,-X_1,-X_2$ repeat cyclically and form a square centered at the origin.