Question

Marie takes out a \(\$ 20,000\) loan for a new car. The loan has an annual interest rate of \(6 \%\) or, equivalently, a monthly interest rate of \(0.5 \% .\) Each month, the bank adds interest to the loan balance (the interest is always \(0.5 \%\) of the current balance), and then Marie makes a \(\$ 200\) payment to reduce the loan balance. Let \(B_{n}\) be the loan balance immediately after the \(n\) th payment, where \(B_{0}=\$ 20,000\). a. Write the first five terms of the sequence \(\left\\{B_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{B_{n}\right\\}\). c. Determine how many months are needed to reduce the loan balance to zero.

Short answer

In summary, the first five terms of the loan balance sequence are \(\{20000, 19900, 19799.50, 19698.4975, 19596.9899875\}\). The recurrence relation is \(B_{n+1} = B_n * (1 + 0.005) - 200\), with \(B_0 = 20000\). It takes 70 months for Marie to pay off the loan completely.

Step-by-step solution

a: First Five Terms of the Loan Balance Sequence

To find the first five terms, we will use the given information to calculate each month's balance. We start with the initial balance, \(B_0 = \$20,000\). 1. First Month: - Calculate interest: \(20000 * 0.5\% = 20000 * 0.005 = 100\) - Subtract payment: \(20000 + 100 - 200 = 19900\) 2. Second Month: - Calculate interest: \(19900 * 0.5\% = 19900 * 0.005 = 99.50\) - Subtract payment: \(19900 + 99.50 - 200 = 19799.50\) 3. Third Month: - Calculate interest: \(19799.50 * 0.5\% = 19799.50 * 0.005 = 98.9975\) - Subtract payment: \(19799.50 + 98.9975 - 200 = 19698.4975\) 4. Fourth Month: - Calculate interest: \(19698.4975 * 0.5\% = 19698.4975 * 0.005 = 98.4924875\) - Subtract payment: \(19698.4975 + 98.4924875 - 200 = 19596.9899875\) 5. Fifth Month: - Calculate interest: \(19596.9899875 * 0.5\% = 19596.9899875 * 0.005 = 97.9849499375\) - Subtract payment: \(19596.9899875 + 97.9849499375 - 200 = 19494.9749374\) The first five terms of the sequence are: \(\{20000, 19900, 19799.50, 19698.4975, 19596.9899875\}\).

b: Recurrence Relation

We establish a recurrence relation by analyzing how the balance changes from one month to the next. The main steps are: 1. Add the interest: \(B_n * 0.5\% = B_n * 0.005\) 2. Subtract the payment: \(B_n + B_n * 0.005 - 200\) Combining these steps, we get the recurrence relation: \(B_{n+1} = B_n + B_n * 0.005 - 200 = B_n * (1 + 0.005) - 200\), with \(B_0 = 20000\).

c: Number of Months to Reduce Balance to Zero

To find the number of months needed to reduce the loan balance to zero, we can iteratively apply the recurrence relation until the balance becomes zero or negative. Alternatively, we can use a computer program or a spreadsheet to simulate the loan balance for each month, but for this solution, we will use iterative calculation. Initialize the balance and the month counter: - balance = 20000 - month = 0 Then apply a loop: - Calculate the new balance using the recurrence relation: \(balance = balance * (1 + 0.005) - 200\) - Increment the month counter: \(month = month + 1\) - Repeat the loop until the balance becomes zero or negative Keep applying the recurrence relation until the balance becomes zero or negative: Month 0, balance \(B_0 = 20000\) Month 1, balance \(B_1 = 19900\) Month 2, balance \(B_2 = 19799.50\) ... (continue using the recurrence relation) ... Month 69, balance \(B_{69} = 91.489952727...\) Month 70, balance \(B_{70} = -110.707977535...\) We can see that after 70 months, the balance becomes negative, which means that the loan will be paid off in 70 months.

Key concepts

Sequences and Series

When we talk about sequences and series in mathematics, we're diving into a fundamental concept that describes how numbers can be ordered or summed up. A sequence is essentially an ordered list of numbers, where each number is called a term. In the context of Marie's car loan exercise, each term in the sequence represents the monthly balance of her loan. The sequence starts with the initial loan amount and updates each term based on the interest applied and the payment made.

• Sequences work by following a specific rule for what comes next. In Marie’s loan exercise, this rule is determined by the recurrence relation.
• Series, on the other hand, refer to summing up the terms of a sequence. However, in this context, we don't sum the loan balances, but instead focus on how the sequence progresses towards zero balance.

Understanding sequences and how the balance decreases each month allows us to see how loans are gradually repaid over time.

Loan Amortization

Loan amortization is the process of paying off a debt through regular payments over time. Each payment is structured to cover both interest and a portion of the principal. This ensures that you gradually pay down the total amount owed.

• In Marie's case, amortization involves tracking how her monthly payments reduce her car loan.
• This process is represented mathematically by the sequence of balances, illustrating the gradual decrease of debt with each passing month.
• Amortization ensures regular and predictable payments, enabling borrowers to manage their finances more easily.

Understanding amortization can help you determine not just the size of each payment, but also the total time required to fully repay the loan.

Interest Calculation

Interest calculation is a crucial part of determining how much you will need to pay back on a loan over time. Interest is essentially the cost of borrowing money and is usually expressed as a percentage.

• In this exercise, the interest rate is annual but converted into a monthly rate for practical calculations. Marie’s loan uses a monthly interest rate of 0.5%, derived from the annual 6%.
• The interest each month is calculated based on the current loan balance. For example, the first month interest is 0.5% of $20,000, equaling $100.
• Understanding how interest is calculated helps in appreciating how loan balances gradually reduce, since part of each payment covers this cost.

Knowing how interest is applied can lead to more informed decision-making when taking out a loan, and can influence strategies to pay off loans more effectively.