Types of Sequences
What is a sequence?
Each number in a sequence is known as a term. We identify a term by its position in the sequence. For example the first term is the term that occurs first in a sequence. The 5th term is the term that occurs in the fifth place of the sequence. The nth term is the term that occurs in the nth position of the sequence.
Arithmetic sequenceIn an arithmetic (linear) sequence the difference between each term is constant. The following sequence is known as an arithmetic sequence
An arithmetic sequence can be expressed in the following recurrence relationship form
Quadratic sequences do not increase in constant amounts. Whenever the second difference is constant in a sequence, the sequence is said to be a quadratic sequence.
The nth term of a Quadratic sequences comes in the form;
A geometric sequence, also known as a geometric progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio
For following sequences are examples of geometric sequences
The general form of a geometric sequence is…
The n-th term of a geometric sequence with initial value a and common ratio r is given by;
The general form of a harmonic sequence is shown below;
A sequence that increases by a constant amount each time is called an arithmetric sequence
The following sequences are all examples of arithmetic sequences;
is an arithmetic progression sequence with common difference 2.
So if you have the arithmetic sequence
where k is the position of a term such as the previous term. This type of sequence where the first term of the sequence and the formula is called an iterative or inductive sequence. It uses an inductive or iterative formulae, this is true for all types of formulas including the geometric sequences. The first term of the sequences and the formula is required. The first term is referred with a1, the second term a2, third term a3 and so forth…a1 = 2 You can go further to generate the following terms in the sequence;
On the other hand you might have a sequence with a deductive or directformulae; For example for our sequence above the formula would be:
where k is the position of the term in question. I have add the +0 to signify the process which must be carried out where we subtract the difference from the first term while forming the formula. The same formula will generate the above sequence except we do not need the first few terms of the sequence to find the following sequences.
You can read further on Arithmetic sequences here.