2025 EJC P1 Q7

Do not use a calculator in answering this question.

An arithmetic sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), \(\ldots \) has \({{u}_{1}}=9\) and \({{u}_{3}}=b\), where \(b\) is a constant.

1. Find \({{u}_{2}}\) in terms of \(b\).[2]

A geometric sequence \({{v}_{1}}\), \({{v}_{2}}\), \({{v}_{3}}\), \(\ldots \) has \({{v}_{1}}=9\) and \({{v}_{3}}=b\).

2. Find the possible values of \({{v}_{2}}\) in terms of \(b\).[2]

It is now given that \({{u}_{2}}-{{v}_{2}}=8\), and the geometric sequence \({{v}_{1}}\), \({{v}_{2}}\), \({{v}_{3}}\), \(\ldots \) is convergent.

3. Find the value of \(b\).
   Hence find \({{u}_{2}}\) and \({{v}_{2}}\).[5]