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A sequence is an infinite list of numbers, like

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

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

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

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

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

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

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

(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}$.

Zvezdelina Stankova-Frenkel 2000-09-20