How much money should be deposited annually in a bank account for 7 years if you wish to withdraw $5000 each year for 3 years, beginning one year after the last​ deposit? The interest rate is 4​% per year.

To solve this problem, we need to determine the annual deposit required in a bank account over 7 years, such that we can withdraw $5000 each year for 3 years, starting one year after the last deposit, given an interest rate of 4% per year.

### Step-by-step solution:

1. **Understanding the Problem:**
   - You will make annual deposits for 7 years.
   - After these deposits, you wish to withdraw $5000 each year for 3 years.
   - The interest rate is 4% per year.

2. **Key Information:**
   - Let $ P $ be the annual deposit amount.
   - Interest rate $ r = 4\% = 0.04 $.
   - Withdrawals: $5000 at the end of years 8, 9, and 10.

3. **Calculate the Present Value of Withdrawals:**
   - The amount needed in the account at the end of year 7 to allow withdrawals of $5000 for 3 years can be determined using the formula for the present value of an ordinary annuity:

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

   where:
   - $ PMT = 5000 $ (the annual withdrawal),
   - $ r = 0.04 $ (the interest rate),
   - $ n = 3 $ (the number of withdrawals).

   Calculating this:

   \[
   PV = 5000 \times \frac{1 - (1 + 0.04)^{-3}}{0.04}
   \]

4. **Compute the Future Value of Annual Deposits:**
   - To reach this present value $ PV $ at the end of 7 years, we need to find the amount $ P $ to be deposited annually.
   - The future value of $ P $ deposited annually for 7 years at 4% interest can be computed using the future value formula of an ordinary annuity:

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

   where:
   - $ P $ is the annual deposit,
   - $ r = 0.04 $,
   - $ n = 7 $.

   The future value $ FV $ should equal the present value $ PV $ calculated in step 3, because this is the amount needed at the end of 7 years.

5. **Set Up the Equation and Solve for $ P $:**

\[
P \times \frac{(1 + 0.04)^7 - 1}{0.04} = 5000 \times \frac{1 - (1 + 0.04)^{-3}}{0.04}
\]

Now let's compute the values and solve for $ P $.The required annual deposit to be made for 7 years is approximately **$1,756.77**.

This amount, when deposited annually in a bank account with a 4% interest rate, will allow you to withdraw $5,000 each year for 3 years, beginning one year after the last deposit.

### Summary:
1. **Annual Deposit Required:** $1,756.77
2. **Total Years of Deposit:** 7 years
3. **Annual Withdrawals:** $5,000 each year for 3 years
4. **Interest Rate:** 4% per year

Would you like more details or have any questions?

### Related Questions:
1. What is the future value of the deposits after 7 years?
2. How does the annual deposit change with a different interest rate?
3. What if the withdrawal amount was $6,000 instead of $5,000?
4. How would the required deposit change if withdrawals start 2 years after the last deposit?
5. How to calculate if there is a monthly deposit instead of an annual one?

### Tip:
Understanding the present value and future value of money is crucial for effective financial planning and investment strategies.

How much money should be deposited annually in a bank account for 7 years if you wish to withdraw $5000 each year for 3 years, beginning one year after the last deposit? The interest rate is 4% per year.

Learn how to calculate the required annual deposits in a bank account to withdraw $5000 each year for 3 years, starting one year after the last deposit, with an interest rate of 4%. Includes step-by-step solution and key formulas for present and future value of annuities.

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