Solve the following problem.

n equals 16n=16​; 

*i*equals 0.053=0.053​; 

PMT equals $ 287PMT=$287​; 

PV equals question markPV=?

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Part 1

PV equals $PV=$enter your response here 

​(Round to two decimal​ places.)




To solve for the present value $ PV $ in an annuity problem, we use the formula for the present value of an annuity:

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

Where:
- $ PMT $ is the payment per period (given as $287),
- $ i $ is the interest rate per period (given as 0.053),
- $ n $ is the number of periods (given as 16).

### Plugging in the values:

\[
PV = 287 \times \frac{1 - (1 + 0.053)^{-16}}{0.053}
\]

Now, let's calculate step by step:

1. **Calculate $ (1 + i) $**:
   \[
   1 + 0.053 = 1.053
   \]

2. **Raise $ 1.053 $ to the power of $-16$**:
   \[
   (1.053)^{-16} \approx 0.4875
   \]

3. **Subtract this result from 1**:
   \[
   1 - 0.4875 = 0.5125
   \]

4. **Divide by the interest rate $ i $**:
   \[
   \frac{0.5125}{0.053} \approx 9.6689
   \]

5. **Multiply by the payment $ PMT $**:
   \[
   PV = 287 \times 9.6689 \approx 2774.00
   \]

### Final Answer:
\[
PV \approx \$2774.00
\]

The present value $ PV $ is approximately **$2774.00**.

Would you like more details or have any other questions?

Here are some related questions:

1. What is the difference between present value and future value?
2. How does changing the interest rate $i$ affect the present value?
3. Can you explain the concept of discounting in relation to present value?
4. What happens to the present value if the number of periods $n$ is increased?
5. How would you calculate the future value of the same annuity?

**Tip**: Always double-check the rounding when dealing with financial calculations to ensure accuracy.

Learn how to calculate the present value (PV) of an annuity with monthly payments (PMT) of $287, an interest rate (i) of 0.053, and 16 periods (n). Follow a step-by-step solution and understand the financial formula used.

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