Problem 56 For $1 \leq n \leq 10,$ find a... [FREE SOLUTION]

Chapter 9: Problem 56

For $1 \leq n \leq 10,$ find a formula for $p_{n},$ the payment in year $n$ on a loan of $\$ 100,000 .$ Interest is $5 \%$ per year, compounded annually, and payments are made at the end of each year for ten years. Each payment is $\$ 10,000$ plus the interest on the amount of money outstanding.

Short Answer

Expert verified

Formula: $ p_n = 10,000 + 0.05 \times (100,000 - 10,000 \times n) $ for $ 1 \leq n \leq 10 $.

Step by step solution

Understand the Loan Conditions

The loan amount is $100,000, and the interest rate is 5% per year, compounded annually. Payments are made at the end of each year for ten years, which includes a fixed payment of $10,000 plus interest on the remaining balance.

Determine the Formula for Annual Payment

Each year's payment includes the fixed principal payment of $10,000 and the interest on the remaining loan balance from the previous year. Thus, the formula for the payment in year $ n $ is $ p_n = 10,000 + 0.05 \times \,\text{remaining balance from year } (n-1) $.

Calculate the Remaining Balance Recursively

To find the remaining balance for year $ n $, subtract the fixed principal payment for each previous year. The balance for the first year is $ 100,000 - 10,000 = 90,000 $. For subsequent years, calculate the balance as follows: remaining balance for year $ n $ = previous year's remaining balance - $ 10,000 $.

Calculate Each Year's Payment

Use the balance from the previous year to calculate each year's payment. For instance, year 1: $ p_1 = 10,000 + 0.05 \times 90,000 = 14,500 $; year 2: calculate new balance 90,000 - 10,000 = 80,000, then $ p_2 = 10,000 + 0.05 \times 80,000 = 14,000 $, and so on.

Generalize the Formula for Any Year

In general, the formula for payment $ p_n $ is $ p_n = 10,000 + 0.05 \times (100,000 - 10,000 \times n) $ for $ n $ from 1 to 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Compound Interest

Compound interest is a powerful concept in finance, where interest is earned not only on the initial principal but also on the accumulated interest from previous periods. In the context of a loan, like the one described, the interest rate is applied to the outstanding balance each year. For our example, the interest rate is 5%. This means each year, 5% interest is added to the remaining balance, which forms part of the total payment. This compounding effect means that interest itself earns interest, making the total amount owed or accumulated grow at an accelerating rate compared to simple interest, where interest is only calculated on the initial principal.

Interest on the remaining balance is calculated annually.
The interest rate for this problem is compounded annually at 5%.
Compound interest ensures that each year, you pay more on interest compared to the linear calculation of simple interest.

Recursive Formula

A recursive formula is a way to find a term in a sequence by referring back to previous terms. In our loan payment example, calculating the payment for year $n$ involves the result from year $n-1$. It follows a simple logic where each year's balance is calculated by reducing the previous year's balance by the fixed principal payment, then adding interest.

The initial balance in year 1 is $100,000 - 10,000 = 90,000$.
The remaining balance for any year $n$ is previous year's remaining balance minus $10,000$.
This recursive pattern makes it easier to calculate each year's payment sequentially.

Fixed Principal Payment

The fixed principal payment refers to the consistent amount that is paid towards the principal balance of the loan every year. In this case, it's $10,000. The principal is the initial amount borrowed, and these payments reduce the principal over time. This fixed component ensures that each year, the loan principal decreases, which impacts the amount of interest accrued in subsequent years.

Every payment includes this principal repayment, reducing the loan balance.
The remaining interest is adjusted annually based on the reduced balance from the fixed principal payment.
In our formula, it simplifies to a deduction of $10,000 from the balance each year.

Annual Payments

Annual payments are the total amount paid at the end of each year, combining the fixed principal payment with the calculated interest on the outstanding balance. For our loan, each payment includes the \(10,000 of principal plus 5% of the prior year's remaining balance. This structure results in a higher initial annual payment as interest is applied to a larger principal, reducing each year as the principal decreases.

Each annual payment consists of \)10,000 and interest from the previous year's outstanding balance.
The pattern of payments changes as the principal reduces consistently, impacting the interest paid each year.
The formula for annual payment $p_n = 10,000 + 0.05 \times (100,000 - 10,000 \times n)$ provides a straightforward calculation for each year.

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Short Answer

Step by step solution

Understand the Loan Conditions

Determine the Formula for Annual Payment

Calculate the Remaining Balance Recursively

Calculate Each Year's Payment

Generalize the Formula for Any Year

Key Concepts

Compound Interest

Recursive Formula

Fixed Principal Payment

Annual Payments

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