Problem 80 - 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 6 months.

Step-by-Step Solution

Answer

The balance after 6 months is approximately $9089.69.

Step 1: Understanding the Recurrence Relation

The recurrence relation provided is a way to model the decreasing balance after each payment. We start with a balance of \(A_0 = 10000\). Each month, the balance \(A_n\) is calculated based on the previous month's balance \(A_{n-1}\), applying the monthly interest of \(0.5\%\) which is a multiplier of \(1.005\) and then subtracting the fixed payment of \(\$200\). Therefore, \(A_n = 1.005 A_{n-1} - 200\).

Step 2: Calculating First Month Balance (\(A_1\))

Using the recurrence relation, compute the balance for the first month: \[ A_1 = 1.005 \times 10000 - 200 = 10050 - 200 = 9850 \]

Step 3: Calculating Second Month Balance (\(A_2\))

Compute the balance after the second month:\[ A_2 = 1.005 \times 9850 - 200 = 9900.25 - 200 = 9700.25 \]

Step 4: Calculating Third Month Balance (\(A_3\))

Compute the balance after the third month:\[ A_3 = 1.005 \times 9700.25 - 200 = 9748.75125 - 200 = 9548.75125 \]

Step 5: Calculating Fourth Month Balance (\(A_4\))

Compute the balance after the fourth month:\[ A_4 = 1.005 \times 9548.75125 - 200 = 9596.49400625 - 200 = 9396.49400625 \]

Step 6: Calculating Fifth Month Balance (\(A_5\))

Compute the balance after the fifth month:\[ A_5 = 1.005 \times 9396.49400625 - 200 = 9443.47647628125 - 200 = 9243.47647628125 \]

Step 7: Calculating Sixth Month Balance (\(A_6\))

Finally, compute the balance after the sixth month:\[ A_6 = 1.005 \times 9243.47647628125 - 200 = 9289.693858662656 - 200 = 9089.693858662656 \]

Key Concepts

Compound Interest

Compound interest is a fundamental concept in finance where the interest on a loan or deposit is calculated based on both the initial principal and the accumulated interest from previous periods. In the context of Margarita's debt, her uncle charges compound interest on her remaining balance each month. This means that Margarita not only pays interest on the original \(\$10,000\) loan but also on any interest that has been added to the balance over time.

• Compound interest enhances the balance if not repaid promptly because it takes into account interest on cumulative interest from previous months.
• In the formula \(A_n = 1.005 A_{n-1} - 200\), the coefficient \(1.005\) represents the additional \(0.5\%\) interest applied each month.

Margarita's balance increases slightly due to compound interest before her monthly payment is applied, thereby elongating the timeline needed to repay the full amount. To manage such loans efficiently, understanding compound interest is crucial, as it helps in planning how to budget repayments effectively.

Loan Repayment

Loan repayment involves paying back a borrowed sum of money over a specified period. In Margarita's case, she has agreed to repay her uncle monthly. Her loan repayment plan requires her to make set payments of \(\$200\) every month.

• Each month's payment includes the interest applied to the remaining balance and reduces the principal amount.
• The effective strategy: make consistent payments on time to avoid incurring additional interest.

In order to repay her debt fully, Margarita must continue with her payments diligently. Any missed or delayed payments could result in more interest being added, complicating her repayment schedule. Having a clear understanding of her loan terms including interest applied and payment due dates will aid in effective repayment management.

Monthly Installments

Monthly installments are regular payments made monthly until a debt is fully paid off. For Margarita, a key feature of her repayment plan is the monthly installment of \(\\(200\). These installments serve two purposes: reducing the principal balance and covering the interest charged on the remaining balance. This setup can be visualized as:

• Interest component: the \(0.5\%\) interest before applying the \(\\)200\) installment.
• Principal reduction: the remaining part of the installment reduces the actual loan amount.

In essence, a set monthly installment assists borrowers by breaking down the repayment to more manageable parts, making it easier to fit into a monthly budget. Borrowers should ensure they can afford these monthly payments to prevent any financial strain.

Interest Calculation

Interest calculation is the process of determining how much interest is added to a principal amount over time. In Margarita's situation, her uncle's method of interest calculation is essential to understanding how the balance evolves each month. Here’s how it works:

• Each month, the balance calculates the interest first using \(0.5\%\) of the outstanding amount. This interest is then added to the current balance.
• Next, Margarita subtracts her monthly payment of \(\$200\), which reduces the new balance, but part of this payment also goes toward the accumulated interest.

By comprehending this calculation, we recognize that the amount of interest paid each month decreases as the balance decreases. This knowledge aids in budgeting for repayments and helps anticipate when the loan will be paid off.