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One common example of exponential growth is population growth. In the simplest model, the growth rate of a population is assumed to be directly proportional to the size of the population at any given time. Let P(t) represent the population at time t, and let r be a constant representing the proportional growth rate. The exponential growth model for population can be represented by the following differential equation: \[P'(t) = rP(t)\] By solving this equation, we arrive at the general formula for exponential growth: \[P(t) = P_0 e^{rt}\] where \(P_0\) is the initial population and e is the base of the natural logarithm (approximately 2.718). This formula shows how the population grows exponentially over time under the given assumptions.

Another example of exponential growth can be found in the concept of compound interest in finance. When money is invested or loaned at a specified interest rate, the interest is often compounded at regular intervals, meaning that the interest amount is added to the principal amount and future interest calculations are based on this new amount. The formula to calculate the future value (A) of an investment with an initial principal (P), an annual interest rate (r), compounded n times per year, and t being the number of years, is given by: \[A = P\left(1 + \frac{r}{n}\right)^{nt}\] This formula demonstrates how the value of the investment grows exponentially over time due to the compounding effect of interest.