2023 H2 Maths Paper 1, Question 11

Approach: the deposits form an arithmetic progression; because the condition is “at least $\$ 50\, 000$” and $a$ is wanted to the nearest cent, we solve the equality first and then check whether rounding up or down is what actually satisfies the inequality.

From 31 January 2021 to 31 December 2023 is exactly 3 years, i.e. $36$ monthly deposits, forming an arithmetic progression with first term $\$ a$, common difference $\$ 50$, and $36$ terms.

Requiring $S_{36} \geq 50\, 000$,

Since $a$ is to the nearest cent, we must round up (rounding down would leave $S_{36} < 50\, 000$).

Approach: set up a recurrence for the amount owed, unroll it for the first few months to see the pattern, then sum the resulting geometric series in general.

Let $L_{n}$ be the amount owed after $n$ months. Each month, interest of $0.1\% = 0.001$ is added first, then the repayment $x$ is subtracted:

Applying this repeatedly,

In general,

The bracketed term is a geometric series of $n$ terms with first term $1$ and common ratio $1.001$:

Substituting back,

as required.  (shown)

For the loan to be fully repaid in $360$ months, $L_{360} = 0$:

Using the rounded repayment from (c)(i), the total repaid over $360$ months is $360 \times \$ 1323.63$. The total interest paid is this total minus the original loan amount:

Approach: with a fixed $\$ 1600$ payment, use the formula from (b) to write $L_{n}$ explicitly, find the (non-integer) $n$ at which $L_{n} = 0$, and note that Wei cannot overpay — so she pays $\$ 1600$ for the largest whole number of months $k$ that still leaves a positive balance, then clears the rest with a smaller final payment $y$.

With $x = 1600$ in the formula from (b),

$L_{n}$ is decreasing (since $(1.001)^{n}$ grows), reaching $0$ when

Since $n$ is not a whole number, Wei makes $k = 287$ full payments of $\$ 1600$, leaving a small positive balance that is cleared by a smaller final payment.

Note: a $288$th payment of $\$ 1600$ would overpay, since $287.83 < 288$.

The final repayment clears this balance after one more month’s interest is added:

The total repaid under the plan in (d)(i):

The total repaid under the plan in (c)(i):