Sequences and exponential functions

An arithmetic sequence adds the same amount d each step: $a_n = a_0 + dn$. A geometric sequence multiplies by the same ratio r each step: $g_n = g_0 r^n$. Arithmetic sequences are linear in n, and geometric sequences are exponential in n.

If you know a term other than the first, start from it: $a_n = a_k + d(n - k)$ and $g_n = g_k r^{n - k}$.

An exponential function has the form $f(x) = ab^x$, with $a \neq 0$, $b > 0$ and $b \neq 1$. The initial value a is $f(0)$. The growth factor b is what the output is multiplied by when x goes up by 1. If $b > 1$ the model grows, and if $0 < b < 1$ it decays.

The key test: over equal steps in x, a linear function adds equal amounts, while an exponential function multiplies by equal factors. If $b = 1 + r$, then r is the percent rate of change per unit of x.

For $a > 0$, $f(x) = ab^x$ is always positive, and the line $y = 0$ is a horizontal asymptote.

To build $f(x) = ab^x$ from two points, divide the outputs. The ratio $\frac{f(x_2)}{f(x_1)}$ equals $b^{x_2 - x_1}$, so take that root to get b. Then use either point to solve for a.

To interpret the parameters, give each one a unit and a moment. Say what a is at the start, then say what happens to the quantity each time the input goes up by 1, as a factor or a percent.