91Ó°ÊÓ

When dealing with loans, interest calculation is crucial, as it determines the additional amount owed above the principal sum. To compute the interest, we multiply the principal by the interest rate, often presented as a percentage. In our example, the initial loan amount is \(100,000, with an annual interest rate of 5%.

This calculation translates to an additional \)5,000 owed at the end of the first year, calculated as follows:

• The formula for computing the interest is: \[\text{Interest} = \text{Principal} \times \text{Interest Rate}.\]
• Substituting our values results in: \[\text{Interest} = 100,000 \times 0.05 = 5,000.\]

It's helpful to view interest payments as the cost of borrowing that remains consistent so long as the interest rate and principal amount remain the same.

Compounded interest refers to the phenomenon where interest is calculated on the initial principal, which then includes all accumulated interest from previous periods. This method increases the total payment over time because each period's interest calculation includes the previous period's interest.

In our scenario, interest is compounded annually. This means each year, the interest is calculated on the remaining balance after the principal payments of $10,000 are deducted. Thus, for subsequent years, the interest charge decreases because the principal being charged diminishes each year by $10,000.
The compounded nature of our interest is straightforward, as it simply applies each year's interest rate to the remaining balance. Therefore, it is slightly different than continuous compounding where interest is calculated more frequently, such as daily or monthly.

A recurrence relation in mathematics is a way of defining a sequence of numbers using previous terms. This is particularly helpful in financial calculations, like loan payments, where the payment amount follows a repeating pattern.

In our loan payment example, we establish a recurrence relation for yearly payments. This involves calculating the payment each year based on the outstanding principal from the previous year.
For year \( n \), the recurrence formula is expressed as:

• \[ p_n = 10,000 + (100,000 - 10,000 \times (n-1)) \times 0.05. \]
• Here, \( p_n \) denotes the payment in year \( n \).

Each year, the term \((100,000 - 10,000 \times (n-1))\) represents the remaining principal on which the interest is calculated. This recurrence captures the gradual decrease in interest owed as the principal reduces annually by the principal payment.