Sequences

A sequence is an ordered list of numbers. A series is the sum of the terms of a sequence.

When describing sequences, the following notation is standard:

$\{a_n\}_{n=1}^{n=10}, \quad a_n = n^2.$

The first part of this description, $\{a_n\}_{n=1}^{n=10}$, could be expanded as a list like this: $a_1, a_2, a_3, \dots, a_9, a_{10}$. It simply means there are ten terms in the sequence with indices ranging from 1 to 10.

The second part, $a_n = n^2$, tells us what each of those terms in the sequence actually is. For example, $a_4 = 4^2 = 16$. Thus,

$\{a_n\}_{n=1}^{n=10}, \quad a_n = n^2 = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.$

A sequence is an ordered set with members called terms.

Usually, the terms are numbers. A sequence can have infinite terms.

An example of a sequence is

$1,2,3,4,5,6,7,8,\dots.$

There are different types of sequences. For example, an arithmetic sequence is when the difference between any two consecutive terms in the sequence is the same. So,

$5, 14, 23, 32, 41,50$

is an arithmetic sequence with common difference $9$, first term $5$, and number of terms $6.$

Another type of sequence is a geometric sequence. This is when the ratio of any two consecutive terms in the sequence is the same. For example,

$2, 6, 18, 54, 162$

is a geometric sequence with common ratio $3$, first term $2$, and number of terms $5.$

In a sequence, it is conventional to use the following variables:

• $a$ is the first term in the sequence.
• $n$ is the number of terms in the sequence.
• ${ T }_{ n }$ is the $n^\text{th}$ term in the sequence.
• ${ S }_{ n }$ is the sum of the first $n$ terms of the sequence.
• $d$ is the common difference between any two consecutive terms (arithmetic sequences only).
• $r$ is the common ratio between any two consecutive terms (geometric sequence only).

For example, if a series starts with $1$ and has a common difference of $1,$ we have ${ S }_{ n }= \dfrac{n(n + 1)}{2}.$
Similarly, for the series of squares $1^2,2^2,3^2,\dots,n^2,$ we have ${ S }_{ n } = \dfrac{n(n + 1)(2n + 1)}{6}.$
For the series of cubes $1^3,2^3,3^3,\dots,n^3,$ we have ${ S }_{ n } = \left(\dfrac{n(n+1)}{2}\right)^2.$

Some special types of sequences can be found in Arithmetic Progressions, Geometric Progressions, and Harmonic Progression.