The formula to calculate the future value $A$ of an investment with daily compounding interest is:

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

Where:

• $P$ is the principal amount (initial investment),
• $r$ is the annual interest rate (in decimal form),
• $n$ is the number of times the interest is compounded per year,
• $t$ is the time the money is invested for (in years),
• $A$ is the future value of the investment.

Given:

• $P = 8000$ (initial investment),
• $r = 0.065$ (annual interest rate of 6.5%),
• $n = 365$ (since interest is compounded daily),
• $t = 7$ years.

Step 1: Substitute the values into the formula

$A = 8000 \left( 1 + \frac{0.065}{365} \right)^{365 \times 7}$

Step 2: Simplify the expression

$A = 8000 \left( 1 + \frac{0.065}{365} \right)^{2555}$ $A = 8000 \left( 1 + 0.000178082 \right)^{2555}$ $A = 8000 \left( 1.000178082 \right)^{2555}$

Step 3: Calculate the exponent

$1.000178082^{2555} \approx 1.5379$

Step 4: Final calculation

$A = 8000 \times 1.5379 \approx 12,303.22$

Conclusion:

The future value of the investment is approximately $12,303.22.

Would you like more details on how the compounding formula works or any further clarifications?

Related Questions:

1. How would the future value change if the interest was compounded quarterly instead of daily?
2. What is the impact of increasing the interest rate on the future value of an investment?
3. How does the time period affect the future value when compounding is daily?
4. How would you calculate the future value if you had monthly contributions to the investment?
5. What is the difference between compound interest and simple interest?

Tip: When dealing with compound interest, increasing the number of compounding periods (e.g., from annually to daily) increases the amount of interest you earn over time, as the interest itself starts earning interest.