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To solve this problem, we will use the formula for future value in compound interest to determine the initial investment $ P $ that Tammy should make.

The formula for compound interest is:

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

Where:
- $ A $ is the future amount Tammy wants to have (here, $ A = 128,900 $).
- $ P $ is the principal amount (initial investment), which we are solving for.
- $ r $ is the annual interest rate (3.45%, or 0.0345 as a decimal).
- $ n $ is the number of times interest is compounded per year (quarterly, so $ n = 4 $).
- $ t $ is the number of years (20 years).

Rearranging to solve for $ P $:

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

### Step-by-Step Solution:

1. **Plug in the values:**

   \[
   P = \frac{128,900}{\left(1 + \frac{0.0345}{4}\right)^{4 \times 20}}
   \]

2. **Calculate the quarterly interest rate:**

   \[
   \frac{0.0345}{4} = 0.008625
   \]

3. **Calculate the exponent:** 

   \[
   4 \times 20 = 80
   \]

4. **Evaluate the compound factor:** 

   \[
   \left(1 + 0.008625\right)^{80}
   \]

5. **Solve for $ P $ after computing the values above.**

Let me calculate this.The initial investment Tammy needs to make is approximately **$64,845**.

Would you like further explanation on any of these steps or additional details?

Here are some related questions you might find helpful:

1. How does changing the interest rate affect the initial investment required?
2. What would happen if the compounding frequency changed from quarterly to monthly?
3. How is the compound interest formula different from simple interest?
4. How long would it take to reach the same amount with a lower interest rate?
5. What if Tammy made additional yearly contributions on top of her initial investment?

**Tip:** When dealing with compounded interest, higher compounding frequencies (e.g., monthly vs. quarterly) result in a larger amount due to the effect of "compounding on compounding."

Tammy has decided to invest to help with her retirement savings. How much would she have to invest to have $128,900 after 20 years, assuming an interest rate of 3.45% compounded quarterly?

Learn how to calculate the initial investment Tammy needs to reach a future value of $128,900 in 20 years, with an interest rate of 3.45% compounded quarterly. This problem covers compound interest, exponential growth, and key financial formulas, suitable for high school students.

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