Derek will deposit $2,541.00 per year for 15.00 years into an account that earns 15.00%. Assuming the first deposit is made 5.00 years from today, how much will be in the account 38.00 years from today?

To solve this problem, we need to calculate the future value of a series of annual deposits made into an account that earns compound interest.

### Problem Breakdown:

1. **Annual Deposit Amount (A):** \$2,541.00
2. **Annual Interest Rate (r):** 15% or 0.15
3. **Number of Deposits (n):** 15 deposits (one each year for 15 years)
4. **First Deposit Starts After (t):** 5 years
5. **Total Time Until Future Value (T):** 38 years

### Step-by-Step Solution:

1. **Determine When Deposits End:**
   - Deposits start 5 years from today.
   - 15 deposits are made over 15 years.
   - Therefore, the last deposit is made at $5 + 14 = 19$ years from today.

2. **Calculate the Future Value of Deposits:**
   - The deposits accumulate interest from the time they are deposited until 38 years from today.
   - The formula for the future value (FV) of an ordinary annuity (where payments are made at the end of each period) is:

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

   - Each deposit will grow from the time it is deposited until 38 years from today. For each deposit, the number of years it grows will be different.

3. **Calculate Future Value for Each Deposit Individually:**

   Since the deposits are made from year 5 to year 19 (15 deposits total), we need to compute the future value for each deposit:

\[
FV_{\text{total}} = \sum_{i=0}^{n-1} A \times (1 + r)^{T - (t + i)}
\]

   Where:
   - $A = 2,541$
   - $r = 0.15$
   - $n = 15$
   - $T = 38$
   - $t = 5$

Let's compute the total future value using the formula above.The total amount in the account 38 years from today will be approximately **$1,720,647.15**.

### Explanation:
Each deposit grows with compound interest from the time it is deposited until 38 years from today. By calculating the future value of each deposit individually and summing them up, we determine the total amount in the account after 38 years.

Would you like more details or have any questions?

### Related Questions:
1. How would the final amount change if the interest rate were different, say 10% or 20%?
2. What if Derek started depositing immediately instead of waiting for 5 years?
3. How much would Derek need to deposit annually to reach a target amount of $2,000,000?
4. How would the future value change if the deposits were made at the beginning of each year instead of the end?
5. What is the effect of compounding frequency (e.g., quarterly vs. annually) on the future value?

### Tip:
For financial problems involving compound interest, understanding the timing of deposits (beginning or end of periods) and the compounding frequency is key to accurate calculations.

Calculate the future value of $2,541 deposited annually at 15% interest, with the first deposit made 5 years from today. The total amount in the account after 38 years will be approximately $1,720,647.15. This problem involves compound interest, ordinary annuities, and financial planning.

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