Questions: How much would you have to invest into a 6 -year certificate of deposit paying 3.8% compounded weekly to make it worth 6500 at the end of the term? Round any intermediate computations. Round your answer to the nearest cent. You would have to invest into the certificate of deposit now.

Solution Steps

To solve this problem, we need to use the formula for compound interest to find the present value (initial investment) that will grow to a specified future value. The formula for compound interest is:

\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Where:

• \( A \) is the future value of the investment ($6500 in this case).
• \( P \) is the present value (the amount to be invested).
• \( r \) is the annual interest rate (3.8% or 0.038).
• \( n \) is the number of times the interest is compounded per year (weekly, so 52 times).
• \( t \) is the number of years the money is invested (6 years).

We need to solve for \( P \):

\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

We are given the following values:

• Future value \( A = 6500 \)
• Annual interest rate \( r = 0.038 \)
• Number of times interest is compounded per year \( n = 52 \)
• Number of years \( t = 6 \)

The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

We need to solve for \( P \): \[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]

Substitute \( A = 6500 \), \( r = 0.038 \), \( n = 52 \), and \( t = 6 \) into the formula: \[ P = \frac{6500}{\left(1 + \frac{0.038}{52}\right)^{52 \times 6}} \]

Perform the calculations step-by-step:

1. Calculate the weekly interest rate: \( \frac{0.038}{52} \approx 0.0007308 \)
2. Add 1 to the weekly interest rate: \( 1 + 0.0007308 \approx 1.0007308 \)
3. Raise the result to the power of \( 52 \times 6 = 312 \): \( (1.0007308)^{312} \approx 1.2558 \)
4. Divide the future value by this result: \( \frac{6500}{1.2558} \approx 5175.24 \)

Final Answer