Sequences

A sequence is an infinite list of numbers, like

$$1, 2, 4, 8, 16, 32, \dots.$$ (1)

The numbers in the sequence are called its terms. The general form of a sequence is

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

where $a_n$ is the $n$-th term of the sequence. In the example (1) above, $a_1=1$, $a_2=2$, $a_3=4$, and so on.

The notations $\{a_n\}$ or $\{a_n\}_{n=1}^\infty$ are abbreviations for

$$a_1, a_2, a_3, \dots.$$

Occasionally the indexing of the terms will start with something other than 1. For example, $\{a_n\}_{n=0}^\infty$ would mean

$$a_0, a_1, a_2, \dots.$$

(In this case $a_n$ would be the $(n+1)$-st term.)

For some sequences, it is possible to give an explicit formula for $a_n$: this means that $a_n$ is expressed as a function of $n$. For instance, the sequence (1) above can be described by the explicit formula $a_n = 2^{n-1}$.