2025 CJC Promo Q4

Aenicillin, a type of bacteria, is grown in a petri dish. Its population doubles every six hours. Denoting the amount of Aenicillin at the beginning of the first day as \(x_0\),

1. write down a recurrence relation for \(x_n\), \(n \geq 1\), the amount of Aenicillin in the petri dish at the end of \(n^{\text{th}}\) day,[1]
2. find \(x_n\), expressing your answer in the form of \(x_n = \mathrm{f}(n)\), \(n \geq 1\).[1]

The growth rate of another bacteria, Benicillin, is known to follow the recurrence relation \(u_n = u_{n-1} + n\), \(n \geq 1\) where \(u_n\) is the amount of Benicillin at the end of \(n^{\text{th}}\) day. The amount of Benicillin in the petri dish at the beginning of the first day, \(u_0\), is 1 unit.

4. Write down \(u_1-u_0\), \(u_2-u_1\) and \(u_3-u_2\). Hence find a quadratic expression for \((u_1-u_0)+(u_2-u_1)+...+(u_{n-1}-u_{n-2})+(u_n-u_{n-1})\) in terms of \(n\).[3]
5. Show that \(\sum_{r=1}^{n}(u_r-u_{r-1})=u_n-u_0\). Using your results in part (c), find \(u_n\) expressing your answer in the form of \(u_n=\mathrm{f}(n)\), \(n \geq 1\).[3]