Page 1: Loan Repayment Calculation

This page introduces a recurrence relation word problem focused on loan repayment calculations. It demonstrates how to use recurrence relations to model financial scenarios, a common topic in Higher Maths and A Level Maths.

The problem presents Betty's loan of £2000 with a 1.5% monthly interest rate and £450 monthly repayments. Students are tasked with creating a recurrence relation to model the loan balance over time.

Definition: A recurrence relation for this loan scenario is given as Bn+1 = 1.015Bn - 450, where Bn represents the balance after n months.

The page walks through the calculation process month by month, showing how the loan balance decreases over time. This step-by-step approach is valuable for students learning to apply recurrence relations to real-world problems.

Example: The calculation shows that by the end of May, the balance owed is £281.82058, and the final payment is made in June.

The problem also asks students to calculate the total amount Betty will have paid, which comes to £2086.05. This illustrates the concept of total interest paid on a loan, an important aspect of financial mathematics.

Highlight: For questions involving exceeding or dropping below specific limits, it's crucial to consider the values both before and after each payment or change.

This page effectively demonstrates the application of recurrence relations in financial contexts, providing a practical example that students might encounter in Higher Maths past papers or A Level Maths exams.

Recurrence Relations - Word Problems

This document covers two main types of recurrence relation word problems: loan repayments and drug dosage calculations. These examples demonstrate the practical applications of recurrence relations in real-world scenarios.

Loan Repayment Problem

The first problem involves a reducing balance loan scenario where Betty borrows £2000 with monthly repayments and interest.

Example: Betty takes out a £2000 loan with 1.5% monthly interest and £450 monthly repayments.

The recurrence relation for the loan balance is derived:

Definition: Bn+1 = 1.015Bn - 450, where Bn is the balance after n months.

The problem is solved step-by-step, showing how the loan balance decreases each month until the final payment in June.

Highlight: The final payment is made in June, with a total repayment of £2086.05.

Drug Dosage Problem

The second problem involves calculating drug levels in a patient's bloodstream using recurrence relations.

Example: A drug with an initial dose of 120ml is administered, with 50% metabolized hourly and 35ml top-ups given each hour.

Two recurrence relations are presented:

1. Un+1 = 0.5Un + 35 (drug level after top-up)
2. Un+1 = 0.5Un (drug level before top-up)

Vocabulary: Metabolised - The process by which a drug is broken down in the body.

The problem is solved to determine when the drug level falls below the effective threshold of 58ml.

Highlight: The treatment stops being effective between the 3rd and 4th doses when the drug level drops below 58ml.