2012 TYS

A sequence \(u_1, u_2, u_3, \dots\) is given by

\[u_1 = 2 \text{ and } u_n = \frac{3u_{n-1} - 1}{6} \text{ for } n \ge 2.\]

1. Find the exact values of \(u_2\) and \(u_3\).
2. It is given that \(u_n \to l\) as \(n \to \infty\). Showing your working, find the exact value of \(l\).
3. For this value of \(l\), verify that \(u_n = \frac{14}{3}\left(\frac{1}{2}\right)^n + l\) satisfies both the initial condition and the recurrence relation given in the question.