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Understanding the concept of an annual interest rate is crucial when dealing with any type of investment or savings account. The annual interest rate, expressed as a percentage, represents the amount a bank or financial institution will pay you for keeping your money with them over the course of a year.

In our example with Samantha, the annual interest rate given is 4.12%. This might initially sound small, but when interest is compounded daily, as in the case of Samantha's deposit, the effects can be quite significant over time. To convert the annual interest rate into a form usable for daily calculations, we divide it by 365, the number of days in a year. This gives us a daily interest rate, which is then applied to the principal amount to calculate the interest earned each day.

When interest is compounded daily, it means that the interest earned each day is added to the principal amount, and the next day's interest calculation is based on this new amount. This essentially results in earning interest on the interest, a powerful concept known as compound interest.

To see how this works step by step, we use the formula for daily compounded interest, which is: \( A = P(1 + \frac{r}{n})^{n*t} \) where \( A \) is the future value of the investment, \( P \) is the principal amount, \( r \) is the annual interest rate in decimal form, \( n \) is the number of times the interest is compounded per year, and \( t \) is the time in years. With Samantha's account being compounded daily, \( n \) is 365. This level of compounding frequency can significantly boost the growth of her investment.

The future value of an investment is simply how much an initial deposit, like Samantha's \$1,500, will be worth in the future after interest has been applied. To find this, we leverage the power of compound interest as it accumulates over time.

Following the given steps in the solution, after converting the annual interest rate to a daily rate, calculating the compounded interest for each day, and finally multiplying by the principal amount, Samantha's future value after three non-leap years is calculated to be approximately \$1,696.62. The detailed formula we used, \( A = P(1 + \frac{r}{n})^{n*t} \) enables us to predict the growth of an investment in scenarios where the interest is compounded at different frequencies, not just daily, making it a versatile tool for financial planning and understanding how savings can grow over time.