Question

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

Plug into the annuity formula

Answer 1

To solve for the future value of an ordinary annuity where $75 is deposited weekly over 2 years at an APR of 4.2% compounded weekly, we first need to identify the parameters for the annuity formula.

### Step 1: Identify \( P \), \( r \), and \( n \)

- **\( P \)**: The amount deposited at each period. In this case, \( P = 75 \) dollars (the weekly contribution).

- **\( r \)**: The interest rate per period. Since the annual interest rate is 4.2%, we convert this to a weekly interest rate. Since there are 52 weeks in a year:
\[
r = \frac{0.042}{52} \approx 0.0008077
\]

- **\( n \)**: The total number of deposits (or compounding periods). Since the investment lasts for 2 years and contributions are made weekly:
\[
n = 2 \times 52 = 104
\]

### Step 2: Plug Into the Annuity Formula

The future value \( FV \) of an ordinary annuity can be calculated using the formula:
\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Substituting the identified values:

1. \( P = 75 \)
2. \( r \approx 0.0008077 \)
3. \( n = 104 \)

Now plug these values into the formula:
\[
FV = 75 \times \frac{(1 + 0.0008077)^{104} - 1}{0.0008077}
\]

### Step 3: Calculate

Let's perform the calculations step by step:

1. Calculate \( (1 + r)^n \):
\[
(1 + 0.0008077)^{104} \approx 1.089138
\]

2. Now calculate \( (1.089138 - 1) \):
\[
1.089138 - 1 \approx 0.089138
\]

3. Now apply to the formula:
\[
FV = 75 \times \frac{0.089138}{0.0008077} \approx 75 \times 110.5535 \approx 8291.51
\]

So the future value of this investment after 2 years is approximately **$8,291.51**.