# Exponential Growth A value is said to grow exponentially over time if it grows proportionally to its current value at all times. Common examples of exponential growth are the growth of the human population (the more people are alive the more children will be born) and growth of investments or debts (compound interest). The general formula for exponential growth of population $P$ at time $t$ is $ P_t = P_0r^t $ where $P_0$ is the initial population size and $r$ the rate of change. If $r$ is bigger than 1 the population will grow, and if it is smaller than 1 it will shrink over time. ## The Compound Interest Formula If the interest of an investment is expressed on a yearly basis (as $r$ per year, say 5% or 0.05), how often the interest is actually paid out will influence the size of the investment after a number of years. The investment can be calculated as such $ A = P\left(1+\frac{r}{n}\right)^{nt} $ where $P$ is the initial value of the investment (or principal in finance terms), $r$ the interest rate per year, $n$ the number of times the interest is paid out in a year, and $t$ the number of year/n intervals that have elapsed. This is a modified version of the general formula for exponential growth. The difference is the meaning of $t$. Notice that if $n=1$, we revert to the general formula.