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 = $
146658

To find the future value ($FV$) of an ordinary annuity, where deposits are made at the end of each period and compounded at the same intervals, we use the following formula:

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

Where:
- $ P $ = Payment per period (in this case, $3,000)
- $ r $ = Interest rate per period (annual rate divided by the number of periods per year)
- $ n $ = Total number of periods (number of deposits over the life of the annuity)

### Given:
- $ P = 3000 $ (quarterly deposit)
- Interest rate $ = 4\% $ per year
- Compounded quarterly means $ r = \frac{0.04}{4} = 0.01 $
- Time = 10 years, so the total number of periods $ n = 10 \times 4 = 40 $

### Plugging into the formula:
\[
FV = 3000 \times \frac{(1 + 0.01)^{40} - 1}{0.01}
\]

Let's calculate this:

1. Calculate $(1 + 0.01)^{40}$:
   \[
   (1 + 0.01)^{40} \approx 1.488864
   \]
2. Subtract 1:
   \[
   1.488864 - 1 = 0.488864
   \]
3. Divide by 0.01:
   \[
   \frac{0.488864}{0.01} = 48.8864
   \]
4. Multiply by $ P $:
   \[
   FV = 3000 \times 48.8864 = 146,659.20
   \]

### Final Answer:
\[
\boxed{146,659.20}
\]

The accumulated future value (rounded to the nearest cent) is \$146,659.20.

Would you like more details or have any questions?

### Follow-up Questions:
1. What would be the future value if the interest rate increased to 5% per year?
2. How would the calculation change if deposits were made monthly instead of quarterly?
3. What is the present value of this annuity given the same conditions?
4. How much interest would be earned over the 10 years with these conditions?
5. What happens to the future value if the deposit period is shortened to 5 years?

### Tip:
Always ensure the compounding frequency matches the deposit frequency when using annuity formulas.

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 = $146,658

Learn how to calculate the future value of an annuity with quarterly deposits of $3,000 over 10 years at 4% interest. This problem covers key financial concepts such as compound interest, annuities, and the time value of money. Step-by-step solution included, suitable for college and advanced high school students.

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