Find the amount accumulated FV in the given annuity account. HINT [See Quick Example 1 and Example 1.] (Assume end-of-period deposits and compounding at the same intervals as deposits. Round your answer to the nearest cent.)
$3,000 is deposited quarterly for 10 years at 4% per year
FV = $

To find the future value (FV) of an ordinary annuity, we use the following formula:

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

Where:
- $ P $ is the periodic payment (deposit),
- $ r $ is the interest rate per period,
- $ n $ is the total number of periods.

### Given:
- $ P = 3,000 $ (quarterly deposit),
- Annual interest rate = 4%,
- Number of years = 10,
- Compounding and deposit frequency = quarterly (4 times a year).

### Step 1: Calculate the interest rate per period ($ r $)
Since the annual interest rate is 4%, and deposits are made quarterly:

\[
r = \frac{4\%}{4} = \frac{0.04}{4} = 0.01
\]

### Step 2: Calculate the total number of periods ($ n $)
Since deposits are made quarterly for 10 years:

\[
n = 4 \times 10 = 40
\]

### Step 3: Plug values into the future value formula

\[
FV = 3,000 \times \left( \frac{(1 + 0.01)^{40} - 1}{0.01} \right)
\]

Now calculate step-by-step:

\[
FV = 3,000 \times \left( \frac{(1.01)^{40} - 1}{0.01} \right)
\]

First, calculate $ (1.01)^{40} $:

\[
1.01^{40} \approx 1.488864
\]

Now, subtract 1:

\[
1.488864 - 1 = 0.488864
\]

Next, divide by 0.01:

\[
\frac{0.488864}{0.01} = 48.8864
\]

Finally, multiply by the periodic payment $ P $:

\[
FV = 3,000 \times 48.8864 = 146,659.20
\]

### Future Value (FV):
\[
FV \approx 146,659.20
\]

Thus, the amount accumulated in the annuity account is **$146,659.20**.

Would you like more details, or have any questions?

### 5 Related Questions:
1. How would the future value change if the interest rate were 5%?
2. What would the future value be if deposits were made monthly instead of quarterly?
3. How is the future value formula different for an annuity due?
4. Can you explain how the interest rate per period is derived in this context?
5. What effect does increasing the deposit amount have on the future value?

### Tip:
Remember that the number of compounding periods directly influences how much interest is earned in an annuity. More frequent compounding generally leads to a higher future value.

Find the amount accumulated FV in the given annuity account. $3,000 is deposited quarterly for 10 years at 4% per year. FV = $

Calculate the future value (FV) of an ordinary annuity with quarterly deposits of $3,000 for 10 years at a 4% annual interest rate. Learn the step-by-step process to solve this finance problem, involving the time value of money, compound interest, and annuities. Perfect for college-level students in finance or business math.

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