To calculate the future value of an investment with compound interest, you can use the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \(A\) is the amount of money accumulated after time \(t\),
- \(P\) is the principal amount (the initial amount of money),
- \(r\) is the annual interest rate (decimal),
- \(n\) is the number of times that interest is compounded per year,
- \(t\) is the number of years the money is invested for.
In this case:
- \(P = 5000\)
- \(r = 0.14\) (14% as a decimal)
- \(n = 12\) (monthly compounding)
- \(t = 15\)
Now, we can plug these values into the formula:
\[ A = 5000 \left(1 + \frac{0.14}{12}\right)^{12 \times 15} \]
First, calculate \(\frac{r}{n}\):
\[ \frac{0.14}{12} \approx 0.01166667 \]
Now calculate \(nt\):
\[ nt = 12 \times 15 = 180 \]
Now we can substitute these values into the formula:
\[ A = 5000 \left(1 + 0.01166667\right)^{180} \]
This simplifies to:
\[ A = 5000 \left(1.01166667\right)^{180} \]
Now we calculate \((1.01166667)^{180}\):
\[ (1.01166667)^{180} \approx 5.188652236 \]
Now calculate \(A\):
\[ A \approx 5000 \times 5.188652236 \approx 25943.26 \]
So, after 15 years, Carl Frederickson will have approximately $25,943.26 in the account.
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