Interest Rates

The formula for calculating the balance after t years for an account with an interest rate of r compounded annually is given by:

\(A = P * (1 + r)^t\)

where A is the balance after t years, P is the initial deposit (or principal), and r is the interest rate expressed as a decimal.

For semi-annual compounding, the interest rate is divided by 2, and the formula becomes:

\(A = P * (1 + (r/2))^(2t)\)

For quarterly compounding, the interest rate is divided by 4, and the formula becomes:

\(A = P * (1 + (r/4))^(4t)\)

For monthly compounding, the interest rate is divided by 12, and the formula becomes:

\(A = P * (1 + (r/12))^(12t)\)

Note that in each case, t represents the number of compounding periods (i.e., years for annual compounding, semi-annual periods for semi-annual compounding, quarterly periods for quarterly compounding, and monthly periods for monthly compounding).

So, in your problem, if Chuck deposits $2000 into a bank account that compounds annually at an interest rate of r, the balance after t years will be:

\(A = 2000 * (1 + r)^t\)

If the interest is compounded semi-annually, the balance after t years will be:

\(A = 2000 * (1 + (r/2))^(2t)\)

If the interest is compounded quarterly, the balance after t years will be:

\(A = 2000 * (1 + (r/4))^(4t)\)

If the interest is compounded monthly, the balance after t years will be:

\(A = 2000 * (1 + (r/12))^(12t)\)