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Suppose [tex]$\$[/tex] 25,000[tex]$ is invested by Jon for 12 years in an account that earns $[/tex]6\%$ interest, compounded quarterly. Round the solutions to the nearest cent, if necessary.

1. Determine the future value of the account.
[tex]\[ \text{Future Value} = \square \][/tex]

2. Determine the amount of interest earned in this account over the 12 years.
[tex]\[ \text{Interest} = \square \][/tex]

Hint: Related Formulas

In the formulas below, [tex]\(A\)[/tex] represents an account balance after [tex]\(t\)[/tex] years, where [tex]\(P\)[/tex] is the principal investment, [tex]\(r\)[/tex] is the annual rate of interest (in decimal form), [tex]\(n\)[/tex] is the number of compounding periods per year, and [tex]\(Y\)[/tex] is the effective annual yield of the investment (in decimal form).
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt} \quad P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \quad A = Pe^{rt} \quad Y = \left(1 + \frac{r}{n}\right)^n - 1
\][/tex]



Answer :

GinnyAnswer
To determine the future value of Jon's investment and the amount of interest earned over the 12-year period, we will use the compound interest formula. Here is the step-by-step solution to the problem:

### Given Information:
- Principal investment ([tex]\(P\)[/tex]) = \[tex]$25,000 - Annual interest rate (\(r\)) = 6% or 0.06 (in decimal form) - Number of compounding periods per year (\(n\)) = 4 (since interest is compounded quarterly) - Number of years (\(t\)) = 12 ### Compound Interest Formula: The future value (\(A\)) of an investment compounded periodically can be determined using the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] ### Step-by-Step Calculation: 1. Convert annual interest rate to the rate per compounding period: \[ \frac{r}{n} = \frac{0.06}{4} = 0.015 \] 2. Calculate the total number of compounding periods: \[ nt = 4 \times 12 = 48 \] 3. Substitute the values into the compound interest formula: \[ A = 25000 \left(1 + 0.015\right)^{48} \] 4. Perform the calculation inside the parentheses: \[ 1 + 0.015 = 1.015 \] 5. Raise 1.015 to the power of 48: \[ 1.015^{48} \approx 2.043478260 \] 6. Multiply the principal by the result: \[ A = 25000 \times 2.043478260 \approx 51086.95652 \] 7. Round to the nearest cent: \[ A \approx 51086.96 \] ### Future Value: \[ \text{Future Value} = \$[/tex]51,086.96 \]

### Amount of Interest Earned:
To find the amount of interest earned over the 12-year period, subtract the initial investment from the future value:
[tex]\[ \text{Interest} = A - P \][/tex]
[tex]\[ \text{Interest} = 51086.96 - 25000 \][/tex]
[tex]\[ \text{Interest} = 26086.96 \][/tex]

### Interest Earned:
[tex]\[ \text{Interest} = \$26,086.96 \][/tex]

### Final Answers:
1. Future Value: \[tex]$51,086.96 2. Interest Earned: \$[/tex]26,086.96

These are the final values of Jon's investment after 12 years with a 6% interest rate compounded quarterly.

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