(a)

Find *An*: *An* = 2,450

1 − (1 + 0.01)−150

0.01

*An* =

 $ 

Identify the following. (Assume the rate and time are using the same time period.)

periodic payment$ ratetime

(b)

Find *R*: 30,000 = *R*

1 − (1 + 0.01)−20

0.01

*R* =

 $ 

Identify the following. (Assume the rate and time are using the same time period.)

present value$ ratetime

Let's break down the problems one by one:

### Part (a)

We are given the following formula:

\[
A_n = \frac{2,450}{\frac{1 - (1 + 0.01)^{-150}}{0.01}}
\]

This formula seems to represent the calculation of the periodic payment for an annuity. We need to simplify the denominator.

#### Step-by-step Solution:
1. The interest rate is $0.01$ (or 1%).
2. The number of periods is $150$.
3. The formula used is for the **present value of an annuity**. In this case, we are solving for the **periodic payment**, given that the present value $A_n = 2450$.

#### Simplification:
The formula can be written as:

\[
A_n = \frac{2450}{\frac{1 - (1.01)^{-150}}{0.01}}
\]

Now, let's simplify this:
\[
1.01^{-150} \approx 0.2236
\]
\[
1 - 0.2236 = 0.7764
\]
\[
\frac{0.7764}{0.01} = 77.64
\]

Finally:
\[
A_n = \frac{2450}{77.64} \approx 31.55
\]

So, the periodic payment $A_n$ is approximately **\$31.55**.

#### Identifying the terms:
- **Periodic Payment**: \$31.55 (from the formula)
- **Present Value (An)**: \$2,450
- **Rate**: 1% (0.01)
- **Time**: 150 periods

---

### Part (b)

We are given:

\[
30,000 = R \cdot \frac{1 - (1 + 0.01)^{-20}}{0.01}
\]

This formula is similar to part (a), but this time we're solving for $R$, which represents the **periodic payment**.

#### Step-by-step Solution:
1. The interest rate is $0.01$ (or 1%).
2. The number of periods is $20$.
3. The formula is for the **present value of an annuity**, and we are solving for $R$ (periodic payment), given that the present value is \$30,000.

#### Simplification:
\[
1.01^{-20} \approx 0.8171
\]
\[
1 - 0.8171 = 0.1829
\]
\[
\frac{0.1829}{0.01} = 18.29
\]

Finally:
\[
R = \frac{30,000}{18.29} \approx 1,640.57
\]

So, the periodic payment $R$ is approximately **\$1,640.57**.

#### Identifying the terms:
- **Present Value**: \$30,000
- **Periodic Payment (R)**: \$1,640.57
- **Rate**: 1% (0.01)
- **Time**: 20 periods

---

### Summary:
- **Part (a)**: The periodic payment $A_n$ is approximately \$31.55.
- **Part (b)**: The periodic payment $R$ is approximately \$1,640.57.

Would you like more details on how these formulas work or explanations on specific steps?

Here are 5 related questions for practice:
1. What would the periodic payment be if the present value was \$5,000 in part (a)?
2. How would the periodic payment change in part (b) if the interest rate was 2% instead of 1%?
3. How do you calculate the future value of an annuity?
4. What is the effect of increasing the number of periods on the periodic payment?
5. How is the present value of a perpetuity calculated?

**Tip**: When dealing with annuities, small changes in the interest rate or the number of periods can have a significant impact on the periodic payment.

Find An: An = 2,450 / (1 - (1 + 0.01)^(-150)) / 0.01. Identify the periodic payment, rate, and time. Also, Find R: 30,000 = R / (1 - (1 + 0.01)^(-20)) / 0.01. Identify the present value, rate, and time.

Learn how to calculate the periodic payment and present value for annuities using the formulas for present value of an annuity. Solve problems involving a $2,450 present value and a $30,000 present value, with interest rates of 1% over 150 and 20 periods, respectively.

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