2024 HCI P2 Q2

A sequence is defined by the recurrence relation

\({{u}_{n+1}}=\frac{1+{{u}_{n}}}{1-{{u}_{n}}}\),

for \(n\ge 1\).

1. State what happens to the sequence when \({{u}_{1}}=0\).[1]

It is now given that \({{u}_{1}}=2\).

2. Find \({{u}_{2}}\), \({{u}_{3}}\), \({{u}_{4}}\), \({{u}_{5}}\), and \({{u}_{6}}\).[2]
3. By observing the pattern in part (b), find \(\displaystyle \underset{r=1}{\overset{4n}{\mathop \sum }}\,{{u}_{r}}\) in terms of \(n\).[2]