Question

"You invest $75 weekly for 2 years at 4.2% APR, compounded weekly." Group 2: Task

Identify P,r,n
A = Future Value of the annuity

P = Periodic Payment

r = Interest rate per period: (APR divided by number of compounding periods per year)

= total number of payments: (years multiplied by compounding periods per year)

Plug into the annuity formula

Show work using Desmos or Google Calculator

Report back to class in a 2-slide presentation (i.e Slides, Canva)

→ How did we adapt the formula for our scenario?

Answer 1

To solve the problem of investing $75 weekly for 2 years at an APR of 4.2% compounded weekly, we need to identify the components for the future value of an annuity formula. Here’s a step-by-step breakdown:

### Step 1: Identify P, r, n

- **P (Periodic Payment)**: This is the amount invested each week.
- \( P = 75 \)

- **r (Interest Rate per Period)**: The annual interest rate needs to be converted to a weekly rate.
- Annual interest rate (APR) = 4.2% = 0.042
- Since the interest is compounded weekly and there are 52 weeks in a year:
- \( r = \frac{0.042}{52} \approx 0.0008076923 \)

- **n (Total Number of Payments)**: This is the total number of weeks in 2 years.
- Total weeks in 2 years = \( 2 \times 52 = 104 \)
- \( n = 104 \)

### Step 2: Future Value of the Annuity Formula

The Future Value (FV) of an annuity formula is given by:
\[
A = P \times \frac{(1 + r)^n - 1}{r}
\]
Where:
- A = Future Value of the annuity
- P = Periodic Payment (in this case, weekly investment of $75)
- r = Interest rate per period (weekly interest)
- n = Total number of payments (weeks)

### Step 3: Plugging in the Values

Substituting the identified values into the formula:
- \( P = 75 \)
- \( r \approx 0.0008076923 \)
- \( n = 104 \)

\[
A = 75 \times \frac{(1 + 0.0008076923)^{104} - 1}{0.0008076923}
\]

### Step 4: Calculate Using a Calculator

Using Desmos or Google Calculator:

1. Calculate \( (1 + 0.0008076923)^{104} \):
- This gives approximately \( 1.089673005 \).

2. Then, calculate:
\[
A = 75 \times \frac{1.089673005 - 1}{0.0008076923}
\]
3. This simplifies to:
\[
A = 75 \times \frac{0.089673005}{0.0008076923} \approx 75 \times 110.336555 \approx 8252.74
\]

### Future Value Result
- \( A \approx 8252.74 \)

### Step 5: Adaptation for Our Scenario

In our scenario, we adapted the standard future value of an annuity formula by:
- Defining the periods to match our investment frequency (weekly).
- Computing a weekly interest rate from the annual percentage rate.
- Calculating the total number of payment periods based on the investment duration.

### Presentation Slides

#### Slide 1: Investment Details
- **Investment Amount**: $75 weekly
- **Investment Duration**: 2 years
- **Annual Interest Rate (APR)**: 4.2%
- **Compounding**: Weekly

#### Slide 2: Future Value Calculation
- **Identified Values**:
- \( P = 75 \)
- \( r = 0.0008076923 \)
- \( n = 104 \)
- **Future Value of Annuity**:
- \( A \approx 8252.74 \)
- **Formula Adaptation Explanation**: Explained how we adjusted the formula for weekly compounding and investment duration.

With this information, you now have an outline for the presentation, the necessary calculations, and a clear understanding of how the formula applies to your scenario.