2025 JPJC P1 Q11

A company posted a video on a social media platform to advertise a new product. The video is uploaded at the start of \(1 ^\mathrm{st}\) April and the number of daily views is recorded at the end of each day. Let \(u_n\), where \(n \geq 1\), denotes the number of daily views recorded each day. It is given that

\[u_{n+1} = (1+k)u_n,\]

where \(k\) is a positive constant.

1. Explain why the sequence \(\{u_n\}\) is a geometric progression.[1]
2. Given that the number of daily views recorded at the end of \(1 ^\mathrm{st}\) April is \(311\) and the total number of daily views recorded from \(1 ^\mathrm{st}\) April to \(3 ^\mathrm{rd}\) April is \(4043\). Find the value of \(k\).[3]
3. Explain why there is no limit to the total number of daily views in the long run.[1]

The company also looks at the number of comments being posted on the social media platform. The number of daily comments is recorded at the end of each day. It is given that \(v_r\), where \(r \geq 1\), denotes the number of daily comments recorded and it is defined by the following relation:

\[v_r = \begin{cases} u_r & \text{for}\hspace{0.5em} 1 \leq r \leq 4, \\ v_{r-1} + 80 & \text{for}\hspace{0.5em} r \geq 5. \end{cases}\]

4. Show that the total number of comments up to the \(r\)th day, where \(r \geq 5\), is \(40r^2 + 8117r - t\), where \(t\) is a constant to be determined.[3]
5. Find the least number of days required for the total number of daily comments to exceed \(100\) \(000\).[3]