2024 EJC P1 Q11

On his \({{45}^{\mathrm{th}}}\) birthday on \(1\) July \(2024\), Mr Eu purchases an annuity plan called EuRetire for a principal sum of \(\$x\). The plan works like this:

Phase 1: The plan pays compound interest at a rate of \(3\%\) per annum on the last day of June each year, until Mr Eu turns \(65\). The total sum at the end of Phase 1 is called the “accumulated sum”.

Phase 2: Upon turning \(65\), Mr Eu receives a monthly payout, \(\$y\), on the \({{1}^{\mathrm{st}}}\) of each month, from the accumulated sum. The amount remaining in the plan continues to draw interest at \(0.2\%\) per month on the last day of each month.

1. Write down an expression for the accumulated sum at the end of Phase 1, giving your answer to \(5\) significant figures.[1]
2. Show that the amount remaining in the plan at the end of the \({{n}^{\mathrm{th}}}\) month of Phase 2, after interest has been credited, is
   \(\$1.8061\left({{1.002}^{n}}\right)x-501\left({{1.002}^{n}}-1\right)y\).[4]
3. Mr Eu wishes to receive \(\$1000\) a month in Phase 2 for at least \(10\) years. What is the minimum principal sum, correct to the nearest dollar, he would need?[2]