91Ó°ÊÓ

Compound interest is calculated on the initial principal amount, as well as any earned interest accrued. This means that the interest earned is not only on the original amount invested but also on any interest that has been added to the investment. The formula for compound interest is given by: \( A = P(1 + \frac{r}{n})^{nt} \) where: A is the future value of the investment P is the initial principal amount r is the annual interest rate (as a decimal) n is the number of times interest is compounded per year t is the number of years

Now that we know the formula, let's analyze its components: 1. \(P(1 + \frac{r}{n})^{nt}\) grows exponentially as time (t) increases, since the exponent "nt" also increases. 2. The part inside the parenthesis \((1 + \frac{r}{n})^{nt}\) represents the growth factor due to compounding interest. It is always greater than 1 for any positive interest rate (r). 3. The growth factor \((1 + \frac{r}{n})^{nt}\) increases as we increase the number of times we compound interest (n).

Now, let's plot the future value of the investment (A) as a function of time (t) for a fixed principal amount (P), interest rate (r), and number of compounding periods (n). 1. The growth factor \((1 + \frac{r}{n})^{nt}\) becomes larger and larger as t increases, which results in an exponential growth of the future value (A). 2. The growth of A will be faster as we increase the compounding periods (n) or the interest rate (r). 3. Since the future value grows exponentially, the graph will have a curve, which further bends upward as time (t) increases, and not a straight line. Therefore, the graph of the future value of a compound interest investment as a function of time is not a straight line but an upward curving exponential graph. This happens because the interest accrues not only on the principal but also on the accumulated interest, leading to a growth that is not constant over time.