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Question

Part 1
A woman decides to invest ​$3000 in an IRA CD at the end of each year for 10 years. If she makes these payments and the certificates all pay 8​%, compounded​ annually, how much will she have at the end of the 10 ​years?
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Part 1
At the end of 10 ​years, the woman will have approximately ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

Answer 1

To calculate the future value of a series of cash flows (payments) made at the end of each year, we can use the formula for the future value of an ordinary annuity:

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

Where:

• \( FV \) is the future value of the annuity.
• \( P \) is the annual payment (in this case, $3000).
• \( r \) is the annual interest rate (in this case, 8% or 0.08).
• \( n \) is the number of payments (in this case, 10 years).

Substituting the values into the formula:

\[ FV = 3000 \times \frac{(1 + 0.08)^{10} - 1}{0.08} \]

First, calculate \( (1 + 0.08)^{10} \):

\[ (1.08)^{10} \approx 2.1589 \]

Now plug this value back into the formula:

\[ FV = 3000 \times \frac{2.1589 - 1}{0.08} \] \[ FV = 3000 \times \frac{1.1589}{0.08} \] \[ FV = 3000 \times 14.48625 \] \[ FV \approx 43458.75 \]

Thus, at the end of 10 years, the woman will have approximately $43,458.75.