Certainly! Let's break down the given exponential expression:
[tex]\[ 5,000\left(1+\frac{0.04}{12}\right)^{12 t} \][/tex]
This expression is used to calculate the future value of an investment that earns interest compounded at regular intervals. Let's identify the components one by one:
1. Principal Amount ([tex]$P$[/tex]): The initial amount invested or principal is [tex]$5,000.
2. Annual Interest Rate ($[/tex]r[tex]$): The annual interest rate in decimal form is 0.04 (which is equivalent to 4%).
3. Compounding Frequency ($[/tex]n[tex]$): This refers to how often the interest is compounded per year. It is represented in the expression by the number of compounding periods per year. This value is 12, indicating monthly compounding.
4. Time ($[/tex]t$): This is the time the money is invested for, in years.
5. Compound Interest Formula: The general formula for compound interest is given by
[tex]\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money).
- [tex]\( r \)[/tex] is the annual interest rate (decimal).
- [tex]\( n \)[/tex] is the number of times that interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for, in years.
In the provided expression:
[tex]\[ 5,000\left(1+\frac{0.04}{12}\right)^{12 t} \][/tex]
- The term [tex]\( \frac{0.04}{12} \)[/tex] represents the interest rate per compounding period.
- The exponent [tex]\( 12t \)[/tex] represents the total number of compounding periods over the entire investment period of [tex]\( t \)[/tex] years.
Therefore, the value that represents the number of times per year that interest is compounded is 12. This value can be found in two places in the expression:
1. In the denominator of the fraction inside the parentheses [tex]\( \frac{0.04}{12} \)[/tex].
2. In the exponent as part of [tex]\( 12t \)[/tex].
