To solve this problem, we can use the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

• $A$ is the amount of money accumulated after interest (including the principal).
• $P$ is the principal amount (initial deposit).
• $r$ is the annual interest rate (decimal form).
• $n$ is the number of times interest is compounded per year.
• $t$ is the time the money is invested or borrowed for, in years.

Given:

• $P = 5000$ (initial deposit)
• $r = 0.07$ (7% annual interest rate, as a decimal)
• $n = 12$ (interest is compounded monthly, so 12 times a year)
• $t = 6$ years

Step 1: Calculate the accumulated amount ($A$) at the end of 6 years.

Substitute the known values into the formula:

$A = 5000 \left(1 + \frac{0.07}{12}\right)^{12 \times 6}$

First, calculate the monthly interest rate and the exponent:

$\frac{0.07}{12} = 0.0058333 \quad \text{(monthly interest rate)}$ $12 \times 6 = 72 \quad \text{(months over 6 years)}$

Now, calculate the total amount:

$A = 5000 \left(1 + 0.0058333\right)^{72}$ $A = 5000 \times (1.0058333)^{72}$ $A \approx 5000 \times 1.48985$ $A \approx 7449.25$

Step 2: Calculate the interest earned.

The total interest earned is the difference between the final amount and the initial principal:

$\text{Interest} = A - P$ $\text{Interest} = 7449.25 - 5000$ $\text{Interest} \approx 2449.25$

Final Answers:

• The amount in the account at the end of 6 years is approximately $7449.25.
• The interest earned during the 6 years is approximately $2449.25.

Would you like more details or have any other questions?

Follow-up Questions:

1. How would the total amount change if the interest were compounded quarterly instead of monthly?
2. What is the formula for continuously compounded interest, and how would it apply to this scenario?
3. How does the interest earned increase as the compounding frequency increases (monthly to daily)?
4. How would the outcome change if the interest rate were higher, say 8% or 10%?
5. How do you calculate the time required for an investment to double using the rule of 72?

Tip: When dealing with compound interest, the more frequently the interest is compounded, the more money you'll accumulate over time!