Use the formula for continuous compounding to compute the balance in the account after​ 1, 5, and 20 years.​ Also, find the APY for the account.
A ​$7000 deposit in an account with an APR of 3.75​%

To solve this, we will use the formula for continuous compounding:

\[
A = P e^{rt}
\]

Where:
- $ A $ is the amount of money in the account after time $ t $,
- $ P $ is the principal (initial deposit),
- $ r $ is the annual interest rate (expressed as a decimal),
- $ t $ is the time the money is invested or borrowed for, in years,
- $ e $ is Euler's number, approximately $ 2.718 $.

### Given:
- $ P = 7000 $,
- $ r = \frac{3.75}{100} = 0.0375 $,
- Time periods: $ t = 1, 5, 20 $ years.

We'll also compute the **Annual Percentage Yield (APY)** for the account, which is given by the formula:

\[
APY = e^r - 1
\]

Let me calculate these values.Here are the results:

- **Balance after 1 year**: \$7,267.48
- **Balance after 5 years**: \$8,443.61
- **Balance after 20 years**: \$14,819.00

The **Annual Percentage Yield (APY)** for the account is approximately **3.82%**.

Would you like more details or have any questions?

Here are five related questions to further explore the topic:
1. How does continuous compounding compare with monthly compounding?
2. What would be the balance after 10 years at the same rate?
3. How does the APY change with different interest rates?
4. How is the APY affected by the frequency of compounding?
5. What is the effect of increasing the initial deposit on the final balance?

**Tip:** The APY gives you a clear picture of the actual interest earned over a year, accounting for compounding effects.

Use the formula for continuous compounding to compute the balance in the account after 1, 5, and 20 years. Also, find the APY for the account. A $7000 deposit in an account with an APR of 3.75%.

Learn how to calculate the future balance using continuous compounding for a $7000 deposit at 3.75% APR over 1, 5, and 20 years. Discover how to compute the Annual Percentage Yield (APY) and understand the impact of exponential growth in finance. Suitable for advanced high school or college students.

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