2024 EJC Promo Q11

A sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), \(\ldots \) is such that \({{u}_{1}}=2\), \({{u}_{n+1}}=k{{u}_{n}}-3\) for all \(n\ge 1\), where \(k\) is a constant.

1. Let \(k=\frac{1}{5}\).
   1. Write down the values of \({{u}_{2}}\), \({{u}_{3}}\) and \({{u}_{4}}\).[2]
   2. Given that the sequence converges to \(l\), find the value of \(l\).[1]
2. Let \(k=1\).
   1. Given that the sequence \({{u}_{1}}\), \({{u}_{2}}\), \({{u}_{3}}\), \(\ldots \) is an arithmetic progression, find the value of \(\sum\limits_{r=1}^{n}{{{u}_{r+4}}}\).[3]
   2. Show that the sequence \({{u}_{7}}\), \({{u}_{10}}\), \({{u}_{13}}\), \({{u}_{16}}\), \(\ldots \), \({{u}_{3n+4}}\), \(\ldots \) is also an arithmetic progression.[2]
   3. Hence, find the value of \(\sum\limits_{r=1}^{100}{{{u}_{3r+4}}}\).[2]