The formula for continuous compounding is given by:

$A = P \cdot e^{rt}$

Where:

• $A$ is the amount of money in the account after time $t$,
• $P$ is the principal amount (initial deposit),
• $r$ is the annual percentage rate (APR) as a decimal,
• $t$ is the time in years,
• $e$ is the base of the natural logarithm (approximately 2.71828).

For this problem:

• $P = 7000$ (initial deposit),
• $r = 4.1\% = 0.041$ (APR),
• $t = 1, 5, 20$ (years).

We will also compute the Annual Percentage Yield (APY), which represents the effective interest rate earned on an account when interest is compounded continuously. The formula for APY is:

$APY = e^r - 1$

Let's calculate both the account balances and the APY.### The account balances after 1, 5, and 20 years are:

• After 1 year: $7,292.96
• After 5 years: $8,592.68
• After 20 years: $15,893.50

The Annual Percentage Yield (APY) is:

• $\text{APY} = 4.19\%$

Would you like further details or have any questions?

Here are some related questions:

1. How does continuous compounding compare to monthly compounding?
2. What effect does increasing the APR have on the APY?
3. What happens to the balance if the time period is doubled?
4. Can you explain the difference between APR and APY in detail?
5. How can we calculate interest for a different compounding period (e.g., quarterly)?

Tip: Continuous compounding yields more than discrete compounding because the interest is being applied constantly.