2009 IJC Promo Q8 (Modified)

A sequence \({{u}_{1}}\), \({{u}_{2}}\) , \({{u}_{3}}\), …. , \({{u}_{n}}\), … is such that \({u_1 = 3}\) and \({{u}_{n+1}}=\frac{3}{4}{{u}_{n}}+4\), for \({ u_{n+1} = \frac{3}{4} u_n + 4, \quad \mathrm{for } n \geq 1}\).

1. Express \({u_n}\) in the form \({16 + A\left(\frac{3}{4}\right)^{n-1}}\), where \({A}\) is a constant to be determined.
2. Determine whether the sequence \({{u}_{1}}\), \({{u}_{2}}\) , \({{u}_{3}}\), …. , \({{u}_{n}}\), … converges.