2022 EJC Promo Q8

A geometric sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), … has a common ratio \(r\), where \(r\) and \({{u}_{1}}\) are positive.

A sequence \({{v}_{1}}\), \({{v}_{2}}\),\({{v}_{3}}\), … is given by \({{v}_{n}}=\ln \left( {{u}_{n}} \right)\) for all positive integer \(n\).

1. Show that \({{v}_{1}}\), \({{v}_{2}}\),\({{v}_{3}}\), … is an arithmetic sequence.[2]

The sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), … is convergent.

2. Find the range of values of the common difference of the sequence \({{v}_{1}}\), \({{v}_{2}}\),\({{v}_{3}}\), …[3]
3. Given that the sum to infinity of the sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), … is \(9\) and \({{v}_{3}}=0\), find the possible values of \(r\) and \({{u}_{1}}\).[4]