A newly launched mobile application tracks the number of active users daily. Let $x_n$ denote

Question

A newly launched mobile application tracks the number of active users daily. Let $x_n$ denote the number of active users at the end of the $n$th day, where $n\ge1$ and let $x_0$ be the number of active users at the beginning of Day 1.

The number of active users triples every day.

Write down a recurrence relation for $x_n$ in terms of $x_{n-1}$, for $n\ge1$.[1]
Find an expression for $x_n$ in terms of $x_0$ and $n$, for $n\ge1$.[2]

To study user retention, a retention index $y_n$ is defined by the recurrence relation

\[y_{n+1}=\frac{4y_n+5}{2y_n+3},\quad n\ge1.\]

Given $y_1=3$, describe the behaviour of the sequence.[1]
Given instead that $y_3=\dfrac{24}{13}$, find the exact value of $y_1$.[2]

Accepted Answer

(a) $x_n=3x_{n-1}$. (b) $x_n=3^nx_0$. (c) Decreases and converges to $1.85$. (d) $y_1=\dfrac12$.

2026 CJC Promo Q4

2026 CJC Promo Q4

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