To solve this, we use the continuous compounding formula:

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

where:

• $A$ is the amount in the account after $t$ years,
• $P$ is the initial principal (in this case, $10,000),
• $r$ is the annual interest rate (APR) in decimal form (3.5% or 0.035),
• $t$ is the time in years,
• $e$ is the base of the natural logarithm, approximately equal to 2.71828.

Calculations:

1. Balance after 1 year ($t = 1$): $A = 10000 \cdot e^{(0.035 \cdot 1)}$

2. Balance after 5 years ($t = 5$): $A = 10000 \cdot e^{(0.035 \cdot 5)}$

3. Balance after 20 years ($t = 20$): $A = 10000 \cdot e^{(0.035 \cdot 20)}$

I'll compute these values and then determine the APY (Annual Percentage Yield).

APY Calculation

The APY can be calculated for continuous compounding as:

$APY = e^r - 1$

Let's proceed with the calculations.Here are the computed balances and APY for the account:

1. Balance after 1 year: $10,356.20
2. Balance after 5 years: $11,912.46
3. Balance after 20 years: $20,137.53

APY (Annual Percentage Yield)

The APY for this account is approximately 3.56%.

Let me know if you’d like further details or explanations.

Related Questions

1. How would the balance change with a higher APR?
2. What if compounding was quarterly instead of continuous?
3. How would the APY differ if compounded annually instead of continuously?
4. What would be the balance after 10 years with the same conditions?
5. How is continuous compounding different from simple interest?

Tip

The APY gives a true annual growth rate, so it’s particularly useful for comparing accounts with different compounding frequencies.