Chapter one of this course taught rates of change. Here is the most consequential one in your adult life: money whose growth rate is proportional to the balance itself. Each year's interest joins the principal and starts earning, so the amount added per year keeps climbing even though the RATE never moves. Simple interest draws a straight line; compound interest draws a curve that steepens forever.

The working form is $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$ — read aloud, A equals P times, one plus r over n, all to the n t. $A$ is the final amount in dollars, $P$ the principal in dollars, $r$ the nominal annual rate as a decimal, $n$ the number of compoundings per year (2 semi-annually, 4 quarterly, 12 monthly), and $t$ the time in years. The letter $n$ does two jobs on purpose: it slices the rate into $r/n$ per period, and it multiplies the exponent to count $nt$ periods. Slice one and forget the other and the answer is quietly wrong.

When interest lands once a period the formula collapses to $A = A_0 (1 + r)^{t}$, where $A_0$ is the starting amount — the same machine with $n = 1$, and the form you will meet everywhere growth is quoted per year.