Determine the term of a 39000 investment with an interest rate of 2.06% compounded quarterly, if the future value is 52000 Select one: a. 16 years b. 17 years c. 15 years d. 14 years

## Step 1: Convert the interest rate to a decimal and per quarter.<br />### The given annual interest rate is $2.06\%$. Convert this to a decimal by dividing by 100, resulting in $0.0206$. Since the interest is compounded quarterly, divide the annual rate by 4 to find the quarterly rate: $0.0206/4 = 0.00515$.<br /><br />## Step 2: Set up the compound interest formula.<br />### The formula for compound interest is $A = P(1 + \frac{r}{n})^{nt}$, where $A$ is the future value, $P$ is the principal, $r$ is the annual interest rate, $n$ is the number of times interest is compounded per year, and $t$ is the time in years. In this case, $A = \$52000$, $P = \$39000$, $r = 0.0206$, and $n = 4$. We want to solve for $t$. The formula becomes $52000 = 39000(1 + \frac{0.0206}{4})^{4t}$.<br /><br />## Step 3: Simplify and solve for $t$.<br />### Simplify the equation: $52000 = 39000(1 + 0.00515)^{4t}$, which becomes $52000 = 39000(1.00515)^{4t}$. Divide both sides by 39000: $\frac{52000}{39000} = (1.00515)^{4t}$, which simplifies to $\frac{52}{39} = \frac{4}{3} = (1.00515)^{4t}$. Take the natural logarithm of both sides: $\ln(\frac{4}{3}) = \ln((1.00515)^{4t})$. Use the logarithm power rule: $\ln(\frac{4}{3}) = 4t \cdot \ln(1.00515)$. Solve for $t$: $t = \frac{\ln(\frac{4}{3})}{4\ln(1.00515)}$. Calculate the value of $t$: $t \approx \frac{0.28768}{4 \cdot 0.005138} \approx \frac{0.28768}{0.020552} \approx 13.998 \approx 14$.