A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant and ${{u}_{n+1}}=\frac{5{{u}_{n}}}{8{

Question

A sequence is such that ${{u}_{1}}=p$, where $p$ is a constant and ${{u}_{n+1}}=\frac{5{{u}_{n}}}{8{{u}_{n}}+1}$, for $n\ge 1$. Describe how the sequence behaves when $p=1$. [2] Find the value of $p$ for which ${{u}_{6}}=\frac{3125}{6253}$. [2] Another sequence ${{v}_{1}}$, ${{v}_{2}}$, ${{v}_{3}}$, … is such that for all $n\ge 1$, ${{v}_{n+2}}-2{{v}_{n+1}}+{{v}_{n}}=k$, where $k$ is a constant. Let ${{w}_{n}}={{v}_{n+1}}-{{v}_{n}}$ for $n\ge 1$. Explain why the sequence $\left\{ {{w}_{n}} \right\}$ is an arithmetic progression. [2]

Accepted Answer

(a)(i) Decreasing and converges to $\frac12$ (a)(ii) $p=\frac15$ (b) $\{w_n\}$ is an AP with common difference $k$

2025 VJC P1 Q4

2025 VJC P1 Q4

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