Problem 74 - Precalculus Mathematics for Calculus

Question

Paying off a Debt Margarita borrows \(\$ 10,000\) from her uncle and agrees to repay it in monthly installments of \(\$ 200\). Her uncle charges \(0.5 \%\) interest per month on the balance. (a) Show that her balance \(A_{n}\) in the \(n\) th month is given recursively by \(A_{0}=10,000\) and $$ A_{n}=1.005 A_{n-1}-200 $$ (b) Find her balance after six months.

Step-by-Step Solution

Answer

The balance after six months is approximately $9,088.67.

Step 1: Understanding the Problem

Margarita borrowed $10,000 and needs to pay it back in monthly installments of $200, with a 0.5% interest per month on the remaining balance. We need to express this with a recursive formula and find the balance after six months.

Step 2: Setting Up the Recursive Formula

Start with the given condition: Margarita's initial balance is \(10,000, so \( A_0 = 10,000 \). Every month, the balance from the previous month is increased by 0.5% (i.e., multiplied by 1.005), and she pays \)200. The recursion formula is:\[ A_n = 1.005 A_{n-1} - 200 \]

Step 3: Calculating Month-by-Month Balances

We'll calculate the balance for six months using the recursive formula:**Month 1:**\[ A_1 = 1.005 imes 10,000 - 200 = 10,050 - 200 = 9,850 \]**Month 2:**\[ A_2 = 1.005 imes 9,850 - 200 = 9,899.25 - 200 = 9,699.25 \]**Month 3:**\[ A_3 = 1.005 imes 9,699.25 - 200 = 9,747.74625 - 200 = 9,547.74625 \]**Month 4:**\[ A_4 = 1.005 imes 9,547.74625 - 200 = 9,595.48498125 - 200 = 9,395.48498125 \]**Month 5:**\[ A_5 = 1.005 imes 9,395.48498125 - 200 = 9,442.46240515625 - 200 = 9,242.46240515625 \]**Month 6:**\[ A_6 = 1.005 imes 9,242.46240515625 - 200 = 9,288.67471718203125 - 200 = 9,088.67471718203125 \]

Step 4: Final Answer

After six months, Margarita's balance is approximately $9,088.67.

Key Concepts

Debt Repayment

Debt repayment is the process of paying back borrowed money over time. In this scenario, Margarita is responsible for repaying a loan of \( \\( 10,000 \) to her uncle. This amount needs to be paid off in installments, which means she pays a portion of it each month.
Each installment is \( \\) 200 \) for Margarita. She pays this amount systematically every month until the debt is fully paid off. This is what's known as an amortizing loan.

• Amortizing Loan: Part of the installment reduces the principal amount, while another part goes to interest.
• Installment: A set amount paid by the borrower regularly.

Understanding debt repayment is crucial as it helps in budgeting personal finances effectively. This helps avoid missed payments and potentially bad credit scores. Managing debt repayment properly can lead to financial stability and freedom.

Interest Calculation

Interest calculation in loans involves determining the extra amount a borrower needs to pay back in addition to the borrowed sum, which is called the principal. For Margarita, her uncle charges a 0.5% interest per month on the outstanding balance of the loan.
This means the remaining amount she owes will increase every month before she makes her installment payment. Calculation of interest can be understood as follows:

• Interest Rate: In this case, 0.5% is applied monthly to the remaining balance.
• Monthly Interest: Calculated by multiplying the remaining balance by 0.005. For example, if the balance is \( \\( 10,000 \), the interest for that month is \( 10,000 \times 0.005 = \\) 50 \).

The calculated interest is then added to the remaining debt balance. Understanding interest is important to grasp how much extra one will pay over the life of a loan. It can significantly impact the total cost of loan repayment.

Financial Mathematics

Financial mathematics is essential for solving problems like Margarita's debt repayment plan using recursive sequences. Financial mathematics provides tools to model and solve real-world monetary problems. In this case, we are using a recursive relation to determine her balance each month.
The recursive formula is given as:\[ A_n = 1.005 A_{n-1} - 200 \] where:

• \( A_n \) is the balance after the \( n^{th} \) month
• \( A_0 \) is the initial balance, \( \$ 10,000 \)
• 1.005 represents the 0.5% interest increase
• \( - 200 \) is the regular payment subtracted each month

Recursive sequences like this one are powerful because they systematically cover multi-step computations without direct monthly calculation. They allow easy adaptation for longer periods and changing conditions, common in complex financial tasks such as savings, investments, or loan repayments.