2.8 Exponential growth and decay  (Page 2/9)

Let’s now turn our attention to a financial application: compound interest . Interest that is not compounded is called simple interest . Simple interest is paid once, at the end of the specified time period (usually $1$ year). So, if we put $\text{\$}1000$ in a savings account earning $2\text{\%}$ simple interest per year, then at the end of the year we have

Compound interest is paid multiple times per year, depending on the compounding period. Therefore, if the bank compounds the interest every $6$ months, it credits half of the year’s interest to the account after $6$ months. During the second half of the year, the account earns interest not only on the initial $\text{\$}1000,$ but also on the interest earned during the first half of the year. Mathematically speaking, at the end of the year, we have

Similarly, if the interest is compounded every $4$ months, we have

and if the interest is compounded daily $(365$ times per year), we have $\text{\$}1020.20.$ If we extend this concept, so that the interest is compounded continuously, after $t$ years we have

Now let’s manipulate this expression so that we have an exponential growth function. Recall that the number $e$ can be expressed as a limit:

Based on this, we want the expression inside the parentheses to have the form $\left(1+1\text{/}m\right).$ Let $n=0.02m.$ Note that as $n\to \infty ,$ $m\to \infty$ as well. Then we get

We recognize the limit inside the brackets as the number $e.$ So, the balance in our bank account after $t$ years is given by $1000e^{0.02t}.$ Generalizing this concept, we see that if a bank account with an initial balance of $\text{\$}P$ earns interest at a rate of $r\text{\%},$ compounded continuously, then the balance of the account after $t$ years is