2009 VJC J1 MYE Q10

The terms of a geometric progression \({{u}_{1}},{{u}_{2}},{{u}_{3}},{{u}_{4}},...\) are such that the sum to infinity is \(81\) and the sum of the first \(4\) terms is \(80\).

If \({{u}_{1}}>100\,\,\)and \(\,\mathrm{n}\ge \mathrm{3}\),

1. find \({{u}_{n}}\) in terms of \(n\),[5]
2. show that \({{u}_{3}}+{{u}_{4}}+{{u}_{5}}+...+{{u}_{n}}=\,p\left[ 1-{{\left( -\frac{1}{3} \right)}^{n+q}} \right]\), where the integers \(p\) and \(q\) are to be found.[3]