- The variable \(n\) in the annuity formula represents the total number of payments.
- Given quarterly payments over 15 years, there are \(4\) payments per year.
- The total number of periods is calculated as \(15 \text{ years} \times 4 \text{ payments/year} = 60\).
- The correct choice is \(\boxed{60}\).
Explanation
- Understanding the Annuity Formula and \(n\)
In the present value of an ordinary annuity formula, \(PV = P \left( \frac{1 - (1 + i)^{-n}}{i} \right)\), the variable \(n\) represents the total number of payment periods over the lifetime of the annuity. The variable \(P\) is the payment amount per period, and \(i\) is the interest rate per period. To find the correct value of \(n\), we need to determine the total number of payments made.
- Calculating the Total Number of Periods \(n\)
The problem states that payments are made quarterly for a period of 15 years. Since a year has 4 quarters, making payments quarterly means there are 4 payments per year. To find the total number of periods \(n\), we multiply the number of years by the number of payment periods per year:
\[n = (\text{number of years}) \times (\text{number of payments per year})\]
\[n = 15 \times 4\]
\[n = 60\]
Thus, there are 60 total payment periods in the 15-year term.
- Selecting the Correct Choice
Comparing our result of \(n = 60\) with the given choices:
We see that our calculated value matches choice b. Therefore, the correct value for \(n\) is \(\boxed{60}\).
Examples
Understanding how to correctly determine the number of payment periods is essential for financial planning. For instance, if you take out a car loan with monthly payments over a 5-year term, you would need to calculate the total number of payments as \(n = 5 \times 12 = 60\). Knowing how to identify these parameters ensures you can accurately use financial formulas to calculate loan repayments, savings goals, or investment growth. Mastering this basic concept is a fundamental skill in personal finance and accounting.