For the sequence \(u_{1}, u_{2}, u_{3}, \ldots\), it is given that \(u_{1}=1\) and \(u_{r+1}=\frac{3 u_{r}-2}{4}\) for all \(r\). Prove by mathematical induction that \(u_{n}=4\left(\frac{3}{4}\right)^{n}-2\), for all positive integers \(n\).