The terms of the sequence $U$ are given by \[u_1 = k \quad \text{and} \quad u_{n+1} = \frac{8u_n -

Question

The terms of the sequence $U$ are given by $$u_1 = k \quad \text{and} \quad u_{n+1} = \frac{8u_n - 14}{u_n - 1}, \quad n \geq 1.$$ For the following values of $k$, describe the behaviour of the sequence $U$. $k = 3$[1] $k = 10$[1] 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 $a \neq \pm 1$. 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 $$w_n = \begin{cases} u_n & \text{when } n \text{ is even,} \ v_n & \text{when } n \text{ is odd.} \end{cases}$$ It is given that the sequence $W$ converges when the sequences $U$ and $V$ converge to the same limit. The sequence $W$ diverges otherwise. 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]

Accepted Answer

(b)k =2 or 7 (c)a: $(-1,0)\cup(0,1)$ and L=$\frac{b}{a}$ (d)b:$(-7,0)\cup(0,7)$

2025 RI P2 Q3

2025 RI P2 Q3