2025 RI P2 Q3

The terms of the sequence \(U\) are given by

\[u_1 = k \quad \text{and} \quad u_{n+1} = \frac{8u_n - 14}{u_n - 1}, \quad n \geq 1.\]

1. For the following values of \(k\), describe the behaviour of the sequence \(U\).
   1. \(k = 3\)[1]
   2. \(k = 10\)[1]
2. Find the possible value(s) of \(k\) if the sequence \(U\) is a constant sequence.[2]

The \(n\)th term of the sequence \(V\) is given by \(v_n = \frac{a^n}{b} + \frac{b}{a(1-a^n)} - \frac{a}{n+1}\), where \(a\) and \(b\) are non-zero real constants and \(a \neq \pm 1\).

3. For some values of \(a\), \(v_n \to L\) as \(n \to \infty\). Find, with justification, the range of values of \(a\) for \(L\) to exist, and state the value of \(L\) in terms of \(a\) and \(b\).[3]

The \(n\)th term of the sequence \(W\) is given by \[w_n = \begin{cases} u_n & \text{when}\hspace{0.5em} n \text{ is even,} \\ v_n & \text{when}\hspace{0.5em} n \text{ is odd.} \end{cases}\]

It is given that the sequence \(W\) converges when the sequences \(U\) and \(V\) converge to the same limit. The sequence \(W\) diverges otherwise.

4. For \(k = 10\), by using part (a)(ii) and part (c), find the range of values of \(b\) for the sequence \(W\) to converge. Hence explain whether \(\sum_{r=1}^{\infty} w_r\) is a convergent series.

   [3]