Question

A woman decides to invest ​$5000 in an IRA CD at the end of each year for 10 years. If she makes these payments and the certificates all pay 7​%, compounded​ annually, how much will she have at the end of the 10 ​years?
Question content area bottom
Part 1
At the end of 10 ​years, the woman will have approximately ​$
  
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 annual investments (an annuity), we can use the formula for the future value of an annuity:

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

Where:
- \( FV \) = future value of the annuity
- \( P \) = payment amount per period
- \( r \) = annual interest rate (as a decimal)
- \( n \) = number of periods (years)

In this case:
- \( P = 5000 \)
- \( r = 0.07 \) (7% as a decimal)
- \( n = 10 \)

Plugging in these values:

\[
FV = 5000 \times \frac{(1 + 0.07)^{10} - 1}{0.07}
\]

Calculating \( (1 + 0.07)^{10} \):

\[
(1 + 0.07)^{10} = 1.967151
\]

Now subtract 1:

\[
1.967151 - 1 = 0.967151
\]

Now, divide by the interest rate:

\[
\frac{0.967151}{0.07} = 13.3878728571
\]

Now multiply by \( P \):

\[
FV = 5000 \times 13.3878728571 \approx 66939.36428571
\]

Finally, rounding to the nearest cent:

\[
FV \approx 66939.36
\]

Thus, at the end of 10 years, the woman will have approximately **$66,939.36**.