Certainly! Let's break down the calculation step by step to determine how much you would have in the account after 21 years.
Firstly, we will use the compound interest formula:
[tex]\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount in the account after the total period.
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested for in years.
Given:
- Principal amount [tex]\( P = \$10,000 \)[/tex]
- Annual interest rate [tex]\( r = 9\% = 0.09 \)[/tex]
- Number of times interest is compounded per year [tex]\( n = 12 \)[/tex]
- Time period [tex]\( t = 21 \)[/tex] years
Now, substituting these values into the formula:
[tex]\[ A = 10000 \left(1 + \frac{0.09}{12}\right)^{12 \times 21} \][/tex]
Let's break this down into smaller steps:
1. Calculate the monthly interest rate: [tex]\( \frac{0.09}{12} \)[/tex]
2. Add 1 to the monthly interest rate.
3. Raise the result to the power of the number of times interest is compounded over the total period, which is [tex]\( 12 \times 21 \)[/tex].
4. Multiply the principal amount by this result.
Finally, the computations should yield:
[tex]\[ A = 10000 \left(1 + 0.0075\right)^{252} \][/tex]
[tex]\[ A = 10000 \left(1.0075\right)^{252} \][/tex]
After computing this, you'll find:
[tex]\[ A = 65728.51 \][/tex]
So, the amount you would have in the account after 21 years, rounded to 2 decimal places, is \$65,728.51.
