A sequence $u_n$ is defined by $u_1=3$, $u_2=1$ and \[u_{n+2}=2u_{n+1}-u_n,\quad\text{for}\hsp

Question

A sequence $u_n$ is defined by $u_1=3$, $u_2=1$ and $$u_{n+2}=2u_{n+1}-u_n,\quad\text{for}\hspace{0.5em}n\in\mathbb{Z}^+.$$ Show that the sequence $u_n$ is an arithmetic progression.[1] A second sequence $v_n$ is defined by $$v_n=k^{u_n},$$ where $k$ is a non-zero constant. Show that the sequence $v_n$ is a geometric progression with common ratio $\dfrac{1}{k^2}$.[2] Find the range of values of $k$ such that $v_n$ is convergent.[2] Evaluate the product of the first $n$ terms of $v_n$, leaving your answer in terms of $k$ and $n$.[3]

Accepted Answer

(a) AP with common difference $-2$ (b) GP with common ratio $\dfrac{1}{k^2}$ (c) $k\leq-1$ or $k\geq1$ (d) $k^{n(4-n)}$.

2026 HCI Promo Q6

2026 HCI Promo Q6

Solution:

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