Question 1
Consider the sequence $(a_n)$ satisfying
$$a_1=\frac12,\qquad a_{n+1}=\sqrt[3]{3a_{n+1}-a_n},\qquad0\le a_n\le1\quad(n\ge1).$$
a) Prove that the sequence is uniquely determined and has a finite limit.
b) Define $b_n=(1+2a_1)(1+2^2a_2)\cdots(1+2^na_n)$ for $n\ge1$. Prove that $(b_n)$ has a finite limit.
$$a_1=\frac12,\qquad a_{n+1}=\sqrt[3]{3a_{n+1}-a_n},\qquad0\le a_n\le1\quad(n\ge1).$$
a) Prove that the sequence is uniquely determined and has a finite limit.
b) Define $b_n=(1+2a_1)(1+2^2a_2)\cdots(1+2^na_n)$ for $n\ge1$. Prove that $(b_n)$ has a finite limit.