A sequence of real numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation \[x_{n+1}=\frac{

Question

A sequence of real numbers $x_1,x_2,x_3,\ldots$ satisfies the recurrence relation $$x_{n+1}=\frac{4x_n+3}{x_n+2}\quad\text{for all}\hspace{0.5em}n\geq1.$$ Given that the sequence converges to $l$, find the possible exact values of $l$.[3] Describe how the sequence behaves when $x_1=2$.[1] Given that $x_4=\frac{187}{63}$, find the value of $x_1$.[3]

Accepted Answer

(a) $l=3$ or $l=-1$. (b) The sequence increases and converges to $3$. (c) $x_1=1$.

Convergence and Reverse Iteration of a Recurrence

Convergence and Reverse Iteration of a Recurrence