Question

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

Identify P,r,n

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 will follow these steps:

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

- **P (payment)**: This is the amount you invest each period. Here, P = $75.
- **r (annual interest rate)**: This is the annual interest rate in decimal form. Since the APR is 4.2%, r = 4.2/100 = 0.042.
- **n (number of compounding periods per year)**: Since the interest is compounded weekly, n = 52 weeks/year.
- **t (total years)**: The total time of investment in years, which is 2 years.

### Step 2: Calculate the total number of compounding periods (N)

The total number of compounding periods over 2 years is:

\[
N = n \times t = 52 \text{ weeks/year} \times 2 \text{ years} = 104 \text{ weeks}
\]

### Step 3: Convert the annual interest rate to the interest rate per period (r/n)

The interest rate per compounding period (weekly) is calculated as:

\[
r/n = \frac{0.042}{52} \approx 0.00080769230769
\]

### Step 4: Plug into the annuity formula

The future value \( FV \) of a series of investments (an annuity) can be calculated using the formula:

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

Substituting the values we have:

\[
FV = 75 \times \left( \frac{(1 + 0.00080769230769)^{104} - 1}{0.00080769230769} \right)
\]

### Step 5: Show work using Google Calculator or Desmos

Using a calculator, we can compute this step by step:

1. **Calculate \( 1 + \frac{r}{n} \)**:
- \( 1 + 0.00080769230769 \approx 1.00080769231 \)

2. **Calculate \( (1 + r/n)^{N} \)**:
- \( (1.00080769231)^{104} \approx 1.090651 \)

3. **Calculate the entire future value formula**:

\[
FV \approx 75 \times \left( \frac{1.090651 - 1}{0.00080769230769} \right)
\]

4. **Calculate:**
- \( 1.090651 - 1 \approx 0.090651 \)
- \( \frac{0.090651}{0.00080769230769} \approx 112.01 \)
- \( FV \approx 75 \times 112.01 \approx 8400.75 \)

### Step 6: Prepare Presentation Slides

**Slide 1: Overview of Investment Calculation**

- **Investment Details**:
- Weekly Investment (P): $75
- Annual Interest Rate (APR): 4.2% (0.042)
- Compounding Frequency (n): Weekly (52 times a year)
- Total Duration (t): 2 years

- **Future Value Calculation**:
- Total compounding periods (N): 104 weeks
- Future Value: ~$8400.75

**Slide 2: Annuity Calculation Adaptation**

- **Formula Used**:
\[
FV = P \times \left( \frac{(1 + r/n)^{N} - 1}{r/n} \right)
\]
- **Adaptation Explanation**:
- We adapted the formula for weekly compounding by dividing the annual interest rate by the number of compounding periods per year and multiplying the investment periods by the number of weeks in 2 years.

### Conclusion

Using this formula and the calculations performed, it is estimated that the total value of your investment after 2 years would be approximately $8400.75. This clearly demonstrates how regular weekly investments can result in a substantial future value when compounded at a manageable interest rate.