A sequence $u_1, u_2, u_3, \dots$ is defined by $$u_{n+1} = 2u_n(1 - u_n)$$ for all positive integ

Question

A sequence $u_1, u_2, u_3, \dots$ is defined by $$u_{n+1} = 2u_n(1 - u_n)$$ for all positive integers $n$. Find the possible limits of the sequence, if the sequence is convergent.[2] For each of the following, evaluate $u_2$ and $u_3$, and state whether the sequence is convergent:[6] $u_1 = 0$, $u_1 = 0.01$, $u_1 = -0.01$.

Accepted Answer

(a) Possible limits $0,\dfrac12$ (b) $u_1=0$: $u_2=u_3=0$, convergent. $u_1=0.01$: $u_2=0.0198,u_3\approx0.0388$, convergent to $\dfrac12$. $u_1=-0.01$: $u_2=-0.0202,u_3\approx-0.0412$, divergent.

2025 EJC P2 Q2

2025 EJC P2 Q2

Solution:

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