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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 { 10 , 20 , 30 , 40 , }

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 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 th term a n can be calculated using the equation a n = 10 n .

Finite and Infinite Sequences

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

Finite sequence: { 3 , 6 , 9 , 12 , , 81 }

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: { 5 , 10 , 15 , 20 , 25 , 30 , }

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    :  1 2 , 1 4 , 1 8 , 1 16 ,

Related infinite series   :  1 2 + 1 4 + 1 8 + 1 16 +

series  can be written in an abbreviated form by using the Greek letter (sigma), called the summation sign.  For instance, to abbreviate the writing of the series 4 + 8 + 12 + 16 + + 100 first notice that the general term of the series is 4 n .  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 n = 1 25 4 n , which is read “the sum of 4 n  for values of  n  from 1 to 25 .”

n = 1 25 4 n = 4 1 + 4 2 + 4 3 + 4 4 + + 4 25

= 4 + 8 + 12 + 16 + + 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 + + 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 +

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.