2025 EJC Promo Q5

The sequence \(x_1, x_2, x_3, \ldots\) satisfies the recurrence relation \[x_{n+1}=x_n^2(2-x_n)\text{ for}\hspace{0.5em}n\geq 1\]

1. Prove algebraically that, if the sequence converges, then it converges to either 1 or 0.[2]
2. Determine the behaviour of the sequence for each of the following cases.
   1. \(x_1=0.5\)[1]
   2. \(x_1=1.5\)[1]
   3. \(x_1=-1\)[1]

3. By considering the graph of \(y=2x^2-x^3-x\), show that if \(0<x_n<1\), then \(x_{n+1}-x_n<0\). State briefly what this result tells us about the sequence in part (b)(i).

   [2]