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 idea is get every term in an arithmetric sequence and divide it by 1. The general form of a harmonic sequence is;
A sequence that increases by a constant amount each time is called an arithmetric sequence The following sequences are all examples of arithmetic sequences. Note that a number is increased by a constant amount known as the common difference.
An arithmetic sequence can be represented in a recurrence relationship form shown below
In the formula k is the position of the previous term. The following arithmetic sequence can be expressed in a recurrence relationship form.
This type of sequence where the first term of the sequence and the formula is required is called an iterative or inductive sequence. The sequence uses an inductive or iterative formulae, this can be true for all types of 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…
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 direct formulae; The sequence in the previous example can also be expressed in a deductive or direct formula.
In this formula k is the position of the term in the sequence. This formula will generate the same sequence except we do not need the first few terms of the sequence to find the entire sequence.
You can learn more about Arithmetic sequences here.
A Geometric sequence is a sequence of numbers where each term in the sequence is found by multiplying the previous term with a with a unchanging number called the common ratio. The following sequence is a geometric sequence;
This geometric sequence has a common ratio 3. Anothe example of a geometric sequence is;
This geometric sequence has a common ratio of ½
In the first the sequence;
Each term in the sequence can be worked out in turn by using the sequence nth term formula. Each term in the sequence is referenced by an where n is the position of the term.
|a1 = 2||a2 = a1 × 2||a3 = a2 × 2||a4 = a3 × 2|
In general we write the formula for the above geometric sequence as:
This type of formula is similar to the formula we saw in the arithmetic sequence so it is a iterative or deductive formula.
You can read more on Geometric progression/sequences here.