1. Periodic Compound Interest: PN = P0 (1 + f )Nk
2. Continuous Compound Interest: PN = P 0 erN
3. Annuity Formula: PN = (\) : where PN is the balance after N years.
d(l-(1+ .!: )-Nk)
4. Payout Annuity: P0 = (f) : where P0 is the balance at the beginning. use these formulas A U.S. citizen invested a $1200 government stimulus check they received on March
L 2020: into stocks that grew with an annual percentage rate of 12 3%: compounded
continuously How much are these stocks worth 6 months later on September 1: 2020?
Round to two decimal place accuracy (Hint: 6 months = 0.5 years)

The formula relevant for this problem is the **Continuous Compound Interest formula**:

\[
P_N = P_0 e^{rN}
\]

Where:
- $ P_N $ is the balance after $ N $ years (the value we want to find),
- $ P_0 $ is the initial investment (in this case, $1200),
- $ r $ is the annual interest rate (in decimal form, so 12.3% becomes $ 0.123 $),
- $ N $ is the time in years (since 6 months = 0.5 years, $ N = 0.5 $).

Now, let's plug the values into the formula:

\[
P_N = 1200 \cdot e^{0.123 \times 0.5}
\]

I'll calculate this next.The value of the stocks after 6 months (0.5 years) is approximately **$1276.12**.

Would you like further details, or do you have any questions about the calculation?

Here are 5 follow-up questions to expand on this topic:
1. What happens to the value of the stocks after 1 year?
2. How would the result change if the interest were compounded periodically instead of continuously?
3. What is the formula for compounding monthly, and how would that apply here?
4. How does the continuous compounding formula compare to simple interest?
5. What is the significance of the number $ e $ in the continuous compounding formula?

**Tip**: When converting an annual interest rate for smaller periods, remember to adjust the time (like using 0.5 years for 6 months) while keeping the interest rate unchanged!

A U.S. citizen invested a $1200 government stimulus check they received on March 1, 2020, into stocks that grew with an annual percentage rate of 12.3%, compounded continuously. How much are these stocks worth 6 months later on September 1, 2020? Round to two decimal place accuracy (Hint: 6 months = 0.5 years).

Learn how to calculate the continuous compound interest on a $1200 investment with a 12.3% APR over 6 months. This example walks through the process of applying the continuous compound interest formula, showcasing how the investment grows to approximately $1276.12.

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