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.