Sequencies - Definitions

The general term of a sequence is the expression that represents all the terms of the sequence (there are usually infinitive terms!). The general term is expressed by an algebraic expression that shows the relation between the value of a certain term in a sequence and the position of that term.
If you have the sequence 2, 8, 14, 20, 26, then each term is 6 more than the previous term. This is an example of an arithmetic progression (AP) and the constant value that defines the difference between any two consecutive terms is called the common difference.

If an arithmetic difference has a first term a and a common difference of d, then we can write a, (a + d), (a + 2d),... {a + (n-1) d} where the nth term = a + (n−1)d

If you have a sequence such as: 81, 27, 9, 3, 1, 1/3, 1/9,... then each term is one third of the term before. 

This can be written as 81, 81(1/3), 81(1/3)2, 81(1/3)3, 81(1/3)4,...

It is an example of a Geometric Progression (GP) where the each term is a multiple of the previous one. The multiplying factor is called the common ratio.

So a GP with a first term a and a common ratio r with n terms, can be stated as
a, ar, ar2, ar3, ar4...arn-1 , where the nth term = arn-1

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