Problem 28 - Algebra 2

Question

LOANS Miguel's monthly car payment is \(\$ 234.85 .\) The recursive formula \(b_{n}=1.005 b_{n}-1-234.85\) describes the balance left on the loan after \(n\) payments. Find the balance of the \(\$ 10,000\) loan after each of the first eight payments.

Step-by-Step Solution

Answer

Balances after payments: 1) $9815.15, 2) $9629.38, 3) $9442.67, 4) $9256.04, 5) $9067.97, 6) $8878.46, 7) $8688.00, 8) $8484.15.

Step 1: Initialization

We start by noting that the initial balance of Miguel's loan (before any payments) is given as \( b_0 = 10000 \), where \( b_0 \) represents the initial loan amount before any payments have been made.

Step 2: Recursive Formula Understanding

The recursive formula given is \( b_n = 1.005 b_{n-1} - 234.85 \). This formula shows us how to find the balance \( b_n \) after \( n \) payments, using the previous balance \( b_{n-1} \). The factor \( 1.005 \) accounts for monthly interest, and \( 234.85 \) is subtracted as the monthly payment.

Step 3: Finding First Month's Balance

To find the balance after the first payment, substitute \( b_0 = 10000 \) into the formula: \[ b_1 = 1.005 \times 10000 - 234.85 = 10050 - 234.85 = 9815.15. \] Thus, the balance after the first payment is \( 9815.15 \).

Step 4: Calculating Balance for Second Payment

Now use \( b_1 \) to find \( b_2 \): \[ b_2 = 1.005 \times 9815.15 - 234.85 = 9864.22575 - 234.85 = 9629.37575. \] So the balance after the second payment is \( 9629.38 \) (rounded to two decimal places).

Step 5: Determining Third Month's Balance

Using the same approach with \( b_2 \): \[ b_3 = 1.005 \times 9629.37575 - 234.85 = 9677.52263 - 234.85 = 9442.67263. \] The balance after the third payment is \( 9442.67 \).

Step 6: Calculate Fourth Payment Balance

Now compute \( b_4 \) using \( b_3 \): \[ b_4 = 1.005 \times 9442.67263 - 234.85 = 9490.88599 - 234.85 = 9256.03599. \] Thus, the balance after the fourth payment is \( 9256.04 \).

Step 7: Finding Fifth Payment Balance

Use \( b_4 \) to determine \( b_5 \): \[ b_5 = 1.005 \times 9256.03599 - 234.85 = 9302.81617 - 234.85 = 9067.96617. \] The balance after the fifth payment is \( 9067.97 \).

Step 8: Calculating Sixth Payment Balance

Substitute \( b_5 \) into the formula: \[ b_6 = 1.005 \times 9067.96617 - 234.85 = 9113.30500 - 234.85 = 8878.45500. \] After the sixth payment, the balance is \( 8878.46 \).

Step 9: Seventh Payment Balance

Calculate the new balance \( b_7 \) using \( b_6 \): \[ b_7 = 1.005 \times 8878.45500 - 234.85 = 8922.84728 - 234.85 = 8687.99728. \] The balance after the seventh payment is \( 8688.00 \).

Step 10: Eighth Payment Balance

Finally, use \( b_7 \) to find \( b_8 \): \[ b_8 = 1.005 \times 8687.99728 - 234.85 = 8731.43727 - 234.85 = 8484.14727. \] Thus, the balance after the eighth payment is \( 8484.15 \).

Key Concepts

Loan Amortization

Loan amortization is a financial concept that involves repaying a loan through regular payments over a set period. Each payment usually consists of both the interest on the outstanding balance and a portion that goes toward reducing the principal, or initial amount borrowed. For example, in the case of Miguel's car loan, his monthly payment of $234.85 consists of both interest and principal repayment.

• With each payment, the outstanding balance decreases, leading to a gradual reduction in the amount of interest paid each month.
• The division between interest and principal in each payment changes over time, with more going toward interest earlier on and more toward the principal later in the loan term.

Miguel’s loan uses a recursive sequence to calculate the balance after each payment, taking into account both the payment amount and the monthly interest rate. This recursive method represents a common approach in loan amortization, as it simplifies the process of tracking the remaining balance after each payment.

Monthly Interest Rate

The monthly interest rate is a key factor in calculating the balance of a loan. It represents the cost of borrowing the remaining loan balance on a monthly basis. The monthly interest rate in Miguel’s loan formula is reflected in the factor 1.005.

• The value 1.005 indicates that the loan balance increases by 0.5% every month due to interest, before accounting for the monthly payment.
• To convert an annual interest rate to a monthly one, divide the annual rate by 12. For example, an annual rate of 6% would correspond to a monthly rate of 0.5%.
• Understanding the monthly interest rate helps in recognizing how much of the monthly payment is applied towards interest and how much reduces the principal balance.

By recognizing the impact of interest on loan balance each month, one can better understand the total cost of borrowing and how quickly the principal is being paid down.

Balance Calculation

Balance calculation for a loan involves determining the remaining amount owed after each payment. In Miguel's case, the balance is found using the recursive formula, where each new balance depends on the previous balance.

• The formula used is: \( b_n = 1.005b_{n-1} - 234.85 \), where \( b_n \) is the balance after \( n \) payments.
• Each step requires calculating the new balance by adding the interest to the previous balance, then subtracting the fixed monthly payment.
• This sequential calculation is repeated for each period until the loan is fully amortized.

By employing this recursive approach, one can efficiently compute the outstanding balance at any point in the loan term, providing insight into how repayments affect the remaining debt. This systematic method ensures accuracy in tracking the reduction of loan principal over time.