Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.)
$55,000 in a fund paying 7% per year, with quarterly payments for 20 years

To solve this problem, we'll use the formula for the future value of an annuity with regular periodic payments:

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

Where:
- $FV$ is the future value (the amount you want to accumulate, $55,000),
- $PMT$ is the periodic payment (which we are solving for),
- $r$ is the interest rate per period,
- $n$ is the total number of periods.

### Step 1: Identify the known values
- $FV = 55,000$
- Annual interest rate = 7%, so quarterly interest rate $r = \frac{7\%}{4} = 0.07 / 4 = 0.0175$
- Number of years = 20, so total number of periods $n = 4 \times 20 = 80$ quarters.

### Step 2: Rearrange the formula to solve for $PMT$

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

### Step 3: Substitute the values into the formula

\[
PMT = \frac{55,000}{\frac{(1 + 0.0175)^{80} - 1}{0.0175}}
\]

### Step 4: Calculate the result

1. Calculate $ (1 + 0.0175)^{80} $:
   \[
   (1 + 0.0175)^{80} \approx 4.3493
   \]

2. Subtract 1:
   \[
   4.3493 - 1 = 3.3493
   \]

3. Divide by $r = 0.0175$:
   \[
   \frac{3.3493}{0.0175} \approx 191.3886
   \]

4. Finally, divide the future value by this result:
   \[
   PMT = \frac{55,000}{191.3886} \approx 287.27
   \]

### Final Answer:
The periodic payment $PMT$ necessary to accumulate $55,000 in the annuity account is approximately **$287.27** per quarter.

Would you like further details or clarifications?

Here are five related questions you may find useful:
1. How does the frequency of compounding affect the future value of an annuity?
2. What happens if the interest rate increases by 1% annually?
3. How do you calculate the present value of an annuity given periodic payments?
4. What is the difference between ordinary annuities and annuities due?
5. How can you calculate the amount accumulated if payments were made monthly instead of quarterly?

**Tip:** The more frequently interest compounds, the greater the accumulation over time!

Find the periodic payments PMT necessary to accumulate the given amount in an annuity account. (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.) $55,000 in a fund paying 7% per year, with quarterly payments for 20 years.

Learn how to calculate the periodic payments necessary to accumulate $55,000 in an annuity account with 7% annual interest and quarterly payments over 20 years. This problem explores the future value of an annuity and compound interest concepts, using step-by-step formulas and explanations.

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