Sequence Notation

A sequence is an ordered list of numbers that goes on forever:

$$a_1,\ a_2,\ a_3,\ \dots,\ a_n,\ \dots$$

$a_n$ is the $n$-th term, and $n$ is its index. The whole sequence is written $\{a_n\}$, or $\{a_n\}_{n=1}^{\infty}$ to show where the index starts.

Two ways to describe one

• A formula: $a_n = \frac{1}{n}$ gives $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$
• A list with a pattern: $2, 4, 6, 8, \dots$ is $a_n = 2n$.

A formula is a function of $n$ whose inputs are only the positive integers. That's why its graph is a row of separate dots, not a curve:

Signs that alternate

A factor of $(-1)^n$ makes the signs alternate, starting negative; $(-1)^{n+1}$ starts positive:

$$\frac{(-1)^{n+1}}{n}:\quad 1,\ -\frac{1}{2},\ \frac{1}{3},\ -\frac{1}{4},\ \dots$$

Finding a formula

Look at how the terms are built: numerators and denominators separately, differences between terms, squares, powers of $2$, factorials.

Terms	Formula
$1, 3, 5, 7, \dots$	$2n - 1$
$1, 4, 9, 16, \dots$	$n^2$
$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \dots$	$\frac{n}{n + 1}$
$2, 4, 8, 16, \dots$	$2^n$
$1, 2, 6, 24, \dots$	$n!$

A few terms never force one formula. $1, 2, 4, \dots$ could continue $8$ (powers of $2$) or $7$ (add $1$, then $2$, then $3$). When we ask for "the" formula, we mean the simplest natural one.

Starting somewhere else

The index can start at $0$ or any other integer. Shifting it changes the formula but not the list: $\{\frac{1}{n + 2}\}_{n=1}^{\infty}$ and $\{\frac{1}{k}\}_{k=3}^{\infty}$ are both $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots$ (substitute $k = n + 2$).