2. For a given value $t$, we consider number sequences $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{n+1}=\frac{a_{n}+t}{a_{n}+1}$ for all $n \geqslant 1$.

(a) Suppose that $t=2$. Determine all starting values $a_{1}>0$ such that $\frac{4}{3} \leqslant a_{n} \leqslant \frac{3}{2}$ holds for all $n \geqslant 2$.

(b) Suppose that $t=-3$. Investigate whether $a_{2020}=a_{1}$ for all starting values $a_{1}$ different from -1 and 1 .