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Sequence Series

A sequence is a function whose domain is the set of positive integers
{ 1, 2, 3, ... }. The functional values or range elements are called the terms of the sequence. A sequence can be defined as follows:

{ an } = a1, a2, a3, ...., an , .... where n = 1, 2, 3, ....
a1, a2, a3, ...., an , .... are terms of the sequence { an }
an = f(n), where f(n) is some function of n, n = 1, 2, 3, ....

A sequence with a first and last term is called a finite sequence, while a sequence with an infinite number of terms is called an infinite sequence.

Associated with any sequence is a series, which is defined as the sum of all the terms of the sequence. Therefore, for the sequence { an } = a1, a2, a3, ...., an , .... , its corresponding series Sn is calculated as

Sn = a1 + a2 + a3 + .... + an + ...

EX. { an } = 1, 3, 5, 7, 9, ..... where an = 2n - 1
Sn = 1 + 3 + 5 + .... + 2n - 1

There are two special kinds of sequences that will be discussed in this section. These are the arithmetic and the geometric sequences (or progressions). An arithmetic sequence is a sequence in which there is a constant difference between successive terms, while a geometric sequence is a sequence wherein each term after the first can be obtained by multiplying the preceding term by a common multiplier. Each of these sequences, along with its associated series, have special properties that will studied here.

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