AP Calculus BC

Arithmetic/Geometric Sequences and Exponential Functions

Growth, decay, or neither

A quantity is exponential when each equal step multiplies the previous value by a constant factor: growth if that factor is greater than 1, decay if it is between 0 and 1. A quantity is linear when each equal step adds a constant amount instead. Anything else, like a ratio of two changing quantities, is neither.

Arithmetic and geometric sequences

An arithmetic sequence has a common difference $d$: each term is the last term plus $d$. A geometric sequence has a common ratio $r$: each term is the last term times $r$.

Given two terms of a sequence that are not next to each other, find $d$ or $r$ by dividing the gap between them across the number of steps: subtract and divide by the number of steps for $d$, or divide and take a root for $r$.

$a_n = a_1 + (n - 1)d \qquad a_n = a_1 \cdot r^{n - 1}$

Compound and continuous interest

$A = P\left(1 + \frac{r}{n}\right)^{nt} \qquad A = Pe^{rt}$

$P$ is the starting amount, $r$ is the annual rate as a decimal, $t$ is time in years, and $n$ is the number of times per year interest compounds. Continuous compounding is the limit as $n \to \infty$, and it always earns slightly more than any finite compounding frequency at the same rate.

Transformations of exponential functions

For $f(x) = a \cdot b^{x - h} + k$, the graph is the parent function $y = b^x$ shifted right $h$ and up $k$. The horizontal asymptote is $y = k$, the domain is all real numbers, and (for $a > 0$) the range is $y > k$. The y-intercept comes from evaluating $f(0)$.