Question

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

Identify P,r,n

Plug into the formula

Show work using Desmos or Google Calculator


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 calculate the future value of an annuity, we can use the following formula:

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

Where:
- \( A \) = Future value of the annuity
- \( P \) = Periodic payment
- \( r \) = Interest rate per period
- \( n \) = Total number of payments

### Given:
1. **Periodic Payment (P):** $75 (weekly contribution)
2. **Annual Interest Rate (APR):** 4.2% or 0.042 (as a decimal)
3. **Number of Years:** 2 years
4. **Compounding Frequency:** Weekly (which means 52 times a year)

### Step 1: Identify Variables
- **P:** $75
- **r:** Annual interest rate divided by the number of compounding periods per year
\[
r = \frac{0.042}{52} = 0.0008076923077 \text{ (approximately)}
\]
- **n:** Total number of payments (compounding periods per year multiplied by the number of years)
\[
n = 52 \times 2 = 104
\]

### Step 2: Plug Values into the Formula
Now, substitute \( P \), \( r \), and \( n \) into the future value formula:

\[
A = 75 \frac{(1 + 0.0008076923077)^{104} - 1}{0.0008076923077}
\]

### Step 3: Calculate Using Desmos or Google Calculator
Now, input this calculation into Desmos or a Google Calculator to find the value of \( A \).

#### Calculation:
1. **Calculate \( (1 + r)^{n} \)**:
\[
(1 + 0.0008076923077)^{104} \approx 1.08570902803
\]

2. **Subtract 1**:
\[
1.08570902803 - 1 \approx 0.08570902803
\]

3. **Divide by \( r \)**:
\[
\frac{0.08570902803}{0.0008076923077} \approx 106.962838009
\]

4. **Multiply by \( P \)**:
\[
A = 75 \times 106.962838009 \approx 8022.228812
\]

### Final Result
The future value of the annuity after 2 years will be approximately \( A \approx 8022.23 \).

### Summary
- **Periodic Payment (P):** $75
- **Interest rate per period (r):** 0.0008076923077 (approx.)
- **Total number of payments (n):** 104
- **Future Value (A):** Approximately $8022.23 after 2 years.