2020 NJC P2 Q3

In the controlled environment of Laboratory (Lab) \(A\) where cells are cultivated, the number of cells increases by \(250\%\) from the start of the week to the end of the week. At the end of each week, \(1.2\) million cells are transferred to another laboratory for experiments. The number of cells (in millions) in Lab \(A\) at the start of the \({{n}^{\mathrm{th}}}\) week is given by \({{a}_{n}}\).

1. Show that \({{a}_{n}}={{3.5}^{n-1}}{{a}_{1}}-0.48\left( {{3.5}^{n-1}}-1 \right)\).[3]

It is given that Lab \(A\) begins operating at the start of the first week with \(0.5\) million cells.

2. Find the number of cells in Lab \(A\) at the end of the third week, before the cells are transferred to another laboratory.[2]

A research requires at least \(60\) million cells before it can commence. To facilitate this, Lab \(A\) and Lab \(B\) are to cultivate and combine all the cells in both the labs by the start of a certain week. Lab B begins operating one week after Lab \(A\).

Lab \(B\) begins operating at the start of that week with \(0.5\) million cells and cultivates \(5\) million cells by the end of that week. At the end of each subsequent week, Lab \(B\) cultivates \(5\) million cells more than in the previous week.

3. In which week can the research first commence?[3]