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.
(i)
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]
(ii)
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 $\$\text{}k$ per annum. A fixed annual bonus of $\$40$ is added to account $B$ at the end of each year of investment.
(iii)
If $k=36$, find the year in which the total amount in account $A$ first exceeds the total amount in account $B$.
[4]
(iv)
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]
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