A woman decides to invest ​$5000 in an IRA CD at the end of each year for 20 years. If she makes these payments and the certificates all pay 6​%, compounded​ annually, how much will she have at the end of the 20 ​years?

Question

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Part 1
At the end of 20 ​years, the woman will have approximately ​$
  
enter your response here.

Answer 1

To calculate the total amount at the end of 20 years for the woman's investment in an IRA CD, we can use the future value of an ordinary annuity formula since she is making regular contributions at the end of each year.

The formula for the future value of an ordinary annuity is:

\[ FV = P \times \left(\frac{(1 + r)^n - 1}{r}\right) \]

Where:

• \( FV \) = future value of the annuity
• \( P \) = payment amount per period (year)
• \( r \) = interest rate per period (annual interest rate)
• \( n \) = number of periods (number of years)

In this case:

• \( P = 5000 \)
• \( r = 0.06 \) (6%)
• \( n = 20 \)

Now we can substitute the values into the formula:

\[ FV = 5000 \times \left(\frac{(1 + 0.06)^{20} - 1}{0.06}\right) \]

Calculating it step by step:

1. Calculate \((1 + 0.06)^{20}\):

\[ (1.06)^{20} \approx 3.207135 \]

2. Now, subtract 1:

\[ 3.207135 - 1 \approx 2.207135 \]

3. Divide by the interest rate \(0.06\):

\[ \frac{2.207135}{0.06} \approx 36.78558 \]

4. Now multiply by the annual payment:

\[ FV = 5000 \times 36.78558 \approx 183,927.90 \]

Therefore, at the end of 20 years, the woman will have approximately:

\[ \text{Amount} \approx 183,927.90 \]

So the final answer is:

$183,927.90