2024 HCI P1 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 \(\sum\limits_{r=1}^{4n}{{{u}_{r}}}\) in terms of \(n\).[2]