N2012 P1 Q3

A sequence \(u_1,u_2,u_3,\ldots\) is given by \(u_1=2\) and \(u_{n+1}=\frac{3u_n-1}{6}\) for \(n\ge1\).

1. Find the exact values of \(u_2\) and \(u_3\).[2]
2. It is given that \(u_n\to l\) as \(n\to\infty\). Showing your working, find the exact value of \(l\).[2]
3. For this value of \(l\), use the method of mathematical induction to prove that \(u_n=\frac{14}{3}(\frac12)^n+l\).[4]