Question 140
For a given value $t$, we consider number sequences $a_1, a_2, a_3, \dots$ such that $a_{n+1} = \frac{a_n + t}{a_n + 1}$ for all $n \ge 1$.
a. Suppose that $t = 2$. Determine all starting values $a_1 \gt 0$ such that $\frac{4}{3} \le a_n \le \frac{3}{2}$ holds for all $n \ge 2$.
b. Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.
a. Suppose that $t = 2$. Determine all starting values $a_1 \gt 0$ such that $\frac{4}{3} \le a_n \le \frac{3}{2}$ holds for all $n \ge 2$.
b. Suppose that $t = -3$. Investigate whether $a_{2020} = a_1$ for all starting values $a_1$ different from $-1$ and $1$.