# Sequence and Series

A sequence is a list of numbers in a certain order. Each number in a sequence is called a term .  Each term in a sequence has a position (first, second, third and so on).

For example, consider the sequence $\left\{10,20,30,40,\dots \right\}$

In the sequence, each number is called a term. The number $10$ has first position, $20$ has second position, $30$ has third position and so on.

The  ${n}^{\text{th}}$ term of a sequence is sometimes written ${a}_{n}$ .

Often, you can find an algebraic expression to represent the relationship between any term in a sequence and its position in the sequence.

In the above sequence, the  ${n}^{\text{th}}$ term ${a}_{n}$ can be calculated using the equation ${a}_{n}=10n$ .

### Finite and Infinite Sequences

A sequence is  finite  if it has a limited number of terms and  infinite  if it does not.

Finite sequence: $\left\{3,6,9,12,\dots ,81\right\}$

The first of the sequence is $3$ and the last term is $81$ . Since the sequence has a last term, it is a finite sequence.

Infinite sequence: $\left\{5,10,15,20,25,30,\dots \right\}$

The first term of the sequence is $5$ . The "..." at the end indicates that the sequence goes on forever; it does not have a last term. It is an infinite sequence.

series   is the indicated sum of the  terms of a sequence.

Example 1:

Finite sequence      : $6,11,16,21,27$

Related finite series      : $6+11+16+21+27$

Infinite sequence    :  $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots$

Related infinite series   :  $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\dots$

series  can be written in an abbreviated form by using the Greek letter $\sum$ (sigma), called the summation sign.  For instance, to abbreviate the writing of the series $4+8+12+16+\cdots +100$ first notice that the general term of the series is $4n$ .  The series begins with the term for  $n=1$ and ends with the term for  $n=25$ .  Using sigma notation, you can write this series as $\underset{n=1}{\overset{25}{\sum }}4n$ , which is read “the sum of $4n$  for values of  $n$  from $1$ to $25$ .”

$\underset{n=1}{\overset{25}{\sum }}4n=4\cdot 1+4\cdot 2+4\cdot 3+4\cdot 4+\cdots +4\cdot 25$

$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=4+8+12+16+\cdots +100$

### Finite and Infinite Series

As in the case of sequences, we can define finite series and infinite series.

A series is  finite  if the corresponding sequence has a limited number of terms and  infinite  if it does not.

Finite series: $7+14+21+28+\cdots +77$

The first of the corresponding sequence is $7$ and the last term is $77$ . Since the corresponding sequence has a last term, it is a finite series.

Infinite series: $7+14+21+28+35+42+\cdots$

The first term of the sequence is $7$ . The "..." at the end indicates that the series goes on forever; it does not have a last term. It is an infinite series.