To find the payment amount for the annuity due, we can use the future value formula for an annuity due:

$FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r)$

Where:

• $FV$ is the future value of the annuity.
• $P$ is the payment amount per period (this is what we need to find).
• $r$ is the interest rate per period.
• $n$ is the total number of periods.

Given:

• Future value, $FV = 14,000$.
• Interest rate per year, $6.8\% = 0.068$.
• Number of years = 17 years.
• Payments are made quarterly, so there are $4$ periods per year.

Steps to calculate:

1. Find the quarterly interest rate:
   $r = \frac{0.068}{4} = 0.017$.

2. Find the total number of periods:
   $n = 17 \times 4 = 68$ quarters.

3. Plug the values into the formula and solve for $P$:

$14,000 = P \times \left( \frac{(1 + 0.017)^{68} - 1}{0.017} \right) \times (1 + 0.017)$

Now, let's compute this:

$14,000 = P \times \left( \frac{(1.017)^{68} - 1}{0.017} \right) \times 1.017$

We will calculate the part inside the parentheses first:

$(1.017)^{68} \approx 3.224$ $3.224 - 1 = 2.224$ $\frac{2.224}{0.017} \approx 130.235$ $130.235 \times 1.017 \approx 132.227$

Now we can solve for $P$:

$14,000 = P \times 132.227$ $P = \frac{14,000}{132.227} \approx 105.28$

Final Answer:

The payment amount should be $105.28.

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Would you like me to explain the steps in more detail or go over anything specific? Here are some related questions:

1. How would the payment change if the interest rate were different?
2. What is the future value formula for an ordinary annuity, and how does it differ from this?
3. How can I calculate the present value of an annuity due?
4. What would happen to the payment amount if the annuity term was shorter or longer?
5. How do I solve for the future value if the payment is already known?

Tip: When working with annuities, always be sure to confirm whether the problem involves an annuity due (where payments are made at the beginning of each period) or an ordinary annuity (where payments are made at the end of each period).