(b) Suppose that Patty's husband delays starting an IRA for the first 10 years he works but then makes $2600 deposits at the end of each of the next 15 years. If the interest rate is 7%, compounded annually, and if he leaves the money in his account for 5 additional years, how much will be in his account at the end of the 30-year period? (Round your answer to the nearest cent.)
$

To calculate the future value of Patty's husband's IRA account at the end of the 30-year period, we need to break the problem into two main parts:

1. **First 15 Years**: Patty's husband makes annual deposits of $2600 at the end of each year for 15 years, earning 7% interest compounded annually.
2. **Next 5 Years**: After the 15 years of contributions, the money remains in the account for 5 more years, continuing to earn 7% interest but without further contributions.

We will use the future value of an annuity formula for the first 15 years of deposits and the future value of a lump sum for the additional 5 years of interest.

### Step 1: Calculate the Future Value of the Annuity (First 15 Years)
The formula for the future value of an annuity compounded annually is:

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

Where:
- $ P = 2600 $ (the annual deposit)
- $ r = 0.07 $ (the annual interest rate)
- $ n = 15 $ (the number of years of contributions)

Substitute the values:

\[
FV_{\text{annuity}} = 2600 \times \frac{(1 + 0.07)^{15} - 1}{0.07}
\]

### Step 2: Allow the Annuity to Grow for 5 More Years (Without Additional Deposits)
After 15 years, the total amount is left to grow for an additional 5 years. The formula for the future value of a lump sum is:

\[
FV_{\text{lump sum}} = FV_{\text{annuity}} \times (1 + r)^t
\]

Where:
- $ FV_{\text{annuity}} $ is the result from Step 1
- $ r = 0.07 $ (the annual interest rate)
- $ t = 5 $ (the number of additional years)

Let's calculate these values step by step.

### Step 1: Future Value of the Annuity
\[
FV_{\text{annuity}} = 2600 \times \frac{(1 + 0.07)^{15} - 1}{0.07} = 2600 \times \frac{(1.07^{15} - 1)}{0.07}
\]

First, calculate $ 1.07^{15} $:

\[
1.07^{15} \approx 2.75903
\]

Now substitute:

\[
FV_{\text{annuity}} = 2600 \times \frac{2.75903 - 1}{0.07} = 2600 \times \frac{1.75903}{0.07} = 2600 \times 25.129
\]

\[
FV_{\text{annuity}} \approx 65335.4
\]

### Step 2: Future Value After 5 Additional Years
Now we calculate the growth of this amount over the next 5 years:

\[
FV_{\text{lump sum}} = 65335.4 \times (1 + 0.07)^5 = 65335.4 \times 1.40255
\]

\[
FV_{\text{lump sum}} \approx 91608.37
\]

### Final Answer:
The amount in the account at the end of the 30-year period is approximately **$91,608.37**.

Would you like more details or further clarifications?

Here are some related questions for deeper exploration:
1. How does the future value change if the interest rate is 8% instead of 7%?
2. What if the contributions were made at the beginning of each year instead of the end?
3. How would the final amount change if the deposits were $3000 instead of $2600?
4. What would be the impact of extending the deposit period to 20 years instead of 15 years?
5. How does the result change if the interest is compounded monthly instead of annually?

**Tip:** The more frequently interest is compounded, the more significant the impact on the future value of an investment.

Suppose that Patty's husband delays starting an IRA for the first 10 years he works but then makes $2600 deposits at the end of each of the next 15 years. If the interest rate is 7%, compounded annually, and if he leaves the money in his account for 5 additional years, how much will be in his account at the end of the 30-year period?

Calculate the future value of an IRA account with $2600 annual deposits for 15 years, compounded at 7% interest. Includes step-by-step breakdown using future value formulas for annuities and lump sums. Learn how the account grows over 30 years.

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