Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 30L / b / Solution

$E[X_{n+1}\mid {\mathcal{F}}_n]=X_n[pc+(1-p)c^{-1}]$. Thus the condition is $pc+(1-p)/c\le 1$, or $p{c}^2-c+1-p\le 0$. Since the roots are $(1-p)/p$ and $1$, the range is $(1-p)/p\le c\le 1$.

Solved by gpt-5.6-sol high.