To compute the balance in the account after 1, 5, and 20 years with continuous compounding, we use the formula for continuous compounding:

$A = P \cdot e^{r \cdot t}$

Where:

• $A$ is the future value (balance in the account),
• $P$ is the principal (initial deposit),
• $r$ is the annual interest rate (as a decimal),
• $t$ is the time in years,
• $e$ is Euler's number (approximately 2.71828).

We are given:

• $P = 8000$ (the initial deposit),
• $r = 0.036$ (3.6% APR expressed as a decimal),
• $t$ is the time in years (1, 5, and 20 years).

1. Compute balance after 1 year

$A_1 = 8000 \cdot e^{0.036 \cdot 1}$

2. Compute balance after 5 years

$A_5 = 8000 \cdot e^{0.036 \cdot 5}$

3. Compute balance after 20 years

$A_{20} = 8000 \cdot e^{0.036 \cdot 20}$

Next, the Annual Percentage Yield (APY) is calculated using the formula:

$APY = e^r - 1$

Let's calculate these values.Here are the results:

1. The balance after 1 year: $8,293.25
2. The balance after 5 years: $9,577.74
3. The balance after 20 years: $16,435.47

The Annual Percentage Yield (APY) for the account is approximately 3.67%.

Would you like more details or have any questions?

Here are some related questions to expand on this topic:

1. How does continuous compounding compare to monthly or quarterly compounding?
2. What would the balance be after 10 years with this interest rate?
3. How does the APY affect the choice of investment accounts?
4. What happens if the APR is increased to 5%? How would that affect the balance after 20 years?
5. Can you explain the difference between APR and APY?

Tip: Continuous compounding grows faster than discrete compounding because interest is added constantly rather than periodically.