2022 TMJC P1 Q11

Mrs Toh intends to invest \(\$1000\) per year in a savings plan, starting from 1 January 2023. Savings plan \(A\) allows her to invest a fixed amount of \(\$1000\) into account \(A\) on the first day of every year. The amount in account \(A\) earns an interest of \(3.5\%\) per annum at the end of each year of investment.

1. Show that the total amount in account \(A\) at the end of \(n\) years is in the form \(p\left( {{q}^{n}}-1 \right)\), where \(p\) and \(q\) are exact constants to be determined.[3]
2. On what date will the total amount in account \(A\) first exceed \(\$36000\)?[3]

Savings plan \(B\) allows Mrs Toh to invest \(\$1000\)into account \(B\) on 1 January 2023. The amount to be invested on the first day of the subsequent years will increase by \(\$\mathrm{}k\) per annum. A fixed annual bonus of \(\$40\) is added to account \(B\) at the end of each year of investment.

3. If \(k=36\), find the year in which the total amount in account \(A\) first exceeds the total amount in account \(B\).[4]
4. Find the least value of \(k\), giving your answer to the nearest whole number, such that the total amount in account \(B\) is more than the total amount in account \(A\) at the end of 31 December 2032.[2]