4. Sarah takes out a mortgage of \(\pounds 155000\) to buy a house. Interest is added each month on the outstanding balance at a constant rate of \(r\) \% each month. Sarah makes fixed monthly repayments to reduce the amount owed.
Each month, interest is added, and then her monthly repayment is used to reduce the outstanding amount owed.
The recurrence relationship for the amount of the mortgage outstanding after \(n + 1\) months is modelled by
$$u _ { n + 1 } = 1.0025 u _ { n } - x \quad n \geqslant 0$$
where \(\pounds u _ { n }\) is the amount of the mortgage outstanding after \(n\) months and \(\pounds x\) is the monthly repayment.
- State the value of \(r\).
- Solve the recurrence relation to find an expression for \(u _ { n }\) in terms of \(x\) and \(n\).
Given that the mortgage will be paid off in exactly 30 years,
- determine, to 2 decimal places, the least possible value of \(x\).
\section*{(Total for Question 4 is 9 marks)}
TOTAL FOR DECISION MATHEMATICS 2 IS 40 MARKS
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