Question

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

Identify P,r,n

←

Plug into the formula

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Show work using Desmos or Google Calculator

→

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

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How did we adapt the formula for our scenario?


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)

Answer 1

To analyze the investment scenario of $75 weekly for 2 years at an APR of 4.2% compounded weekly, we can follow these steps:

### 1. Identify P, r, n

**P** = Periodic Payment
- In this case, \( P = 75 \) (the amount invested each week).

**r** = Interest Rate per Period
- The Annual Percentage Rate (APR) is given as 4.2%. Since it's compounded weekly, we need to convert this to a weekly rate:
\[
r = \frac{0.042}{52} \approx 0.00080769
\]

**n** = Total Number of Payments
- The investment period is 2 years, and since there are 52 weeks in a year, we find:
\[
n = 2 \times 52 = 104
\]

### 2. Plug into the formula

For an annuity compounded periodically, the formula for the future value \( A \) is given by:

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

Substituting the values we identified into the formula:

\[
A = 75 \times \frac{(1 + 0.00080769)^{104} - 1}{0.00080769}
\]

### 3. Show Work Using a Calculator

Using Desmos or Google Calculator, you can compute \( A \):
1. Calculate \( (1 + r)^n \):
- \( (1 + 0.00080769)^{104} \) computes to approximately **1.09068959**.

2. Subtract 1 from this result:
- \( 1.09068959 - 1 = 0.09068959 \).

3. Now, calculate the entire equation:
- \( A = 75 \times \frac{0.09068959}{0.00080769} \approx 75 \times 112.50951 \approx 8438.21 \).

Thus, the future value of the annuity after 2 years is approximately **$8438.21**.

### 4. Presentation Slides Outline

#### Slide 1: Investment Overview
- Title: Investment Analysis
- Content:
- Investment amount: $75 weekly for 2 years
- Interest Rate: 4.2% APR, compounded weekly
- Components identified:
- \( P = 75 \)
- \( r \approx 0.00080769 \)
- \( n = 104 \)

#### Slide 2: Future Value of the Annuity
- Title: Future Value Calculation
- Content:
- Future Value Formula: \( A = P \times \frac{(1 + r)^n - 1}{r} \)
- Calculation in steps:
- \( (1 + r)^n \approx 1.09068959 \)
- Resulting in \( A \approx 8438.21 \)
- Conclusion: The future value of the investment is approximately **$8438.21**.

### 5. How did we adapt the formula for our scenario?

In our scenario:
- We adapted the future value of an ordinary annuity formula to accommodate for compounding interest on a weekly basis.
- By determining the weekly interest rate and the total number of weekly payments over 2 years, we used these metrics to correctly calculate the future value of the weekly investments.

This presentation structure and calculations provide a comprehensive analysis of the investment scenario using the adapted formula tailored to the specifics of compounding frequency and payment intervals.