# Sequence

Topics: Sequence, Natural number, Series Pages: 3 (654 words) Published: March 7, 2013
SEQUENCE
* In mathematics, informally speaking, a sequence is an ordered list of objects (or events). Like a set, it contains members (also called elements, or terms). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers. * For example, {M, A, R, Y} is a sequence of letters with the letter 'M' first and 'Y' last. This sequence differs from {A, R, M, Y}. Also, the sequence {1, 1, 2, 3, 5, 8}, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers {2, 4, 6,...}. Finite sequences are sometimes known as strings or words and infinite sequences as streams. The empty sequence { } is included in most notions of sequence, but may be excluded depending on the context. ARITHMETIC SEQUENCE

* A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n.

Explicit Formula:| an = a1 + (n – 1)d|
Example 1:| 3, 7, 11, 15, 19 has a1 = 3, d = 4,
and n = 5. The explicit formula is
an = 3 + (n – 1)·4 = 4n – 1|
Example 2:| 3, –2, –7, –12 has a1 = 3, d = –5, and n = 4. The explicit formula is
an = 3 + (n – 1)(–5) = 8 – 5n|

GEOMETRIC SEQUENCE
* Before talking about geometric sequence, in math, a sequence is a set of numbers that follow a pattern. We call each number in the sequence a term.

For examples, the following are sequences:

2, 4, 8, 16, 32, 64, .......

243, 81, 27, 9, 3, 1, ...............

A geometric sequence is a sequence where each term is found by multiplying or dividing the same...