# 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 sequence

In an arithmetic (linear) sequence the difference between each term is constant. The following sequence is known as an arithmetic sequenceAn arithmetic sequence can be expressed in the following recurrence relationship form

_{k + 1}= U

_{k}+ n

### Quadratic Sequence

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;

### Geometric Sequence

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;

_{n}= ar

^{n-1}

## Harmonic Sequence

The general form of a harmonic sequence is shown below;

## Arithmetic progression/sequences

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

_{k+1}= a

_{k}+2

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 a_{1}, the second term a_{2}, third term a_{3} and so forth…

_{1}= 2

*You can go further to generate the following terms in the sequence;*

_{2}= a

_{1}+2 a

_{3}= a

_{2}+2 a

_{4}= a

_{3}+2 a

_{5}= a

_{4}+2

On the other hand you might have a sequence with a deductive or directformulae; For example for our sequence above the formula would be:

_{k}= 2k+0

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**.

## Geometric progression/sequences

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. For instance, the sequence 2, 6, 18, 54, … is a geometric sequence with common ratio 3. Similarly with 10, 5, 2.5, 1.25, … which is a geometric sequence with common ratio1/2 In the first instance the sequence_{1}= 2 a

_{2}= a

_{1}× 2 a

_{3}= a

_{2}× 2 a

_{4}= a

_{3}× 2

_{k+1}= a

_{k}× 3

**1**will give you the position of that term you’re after, usually that number will be the previous number. 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**.

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hii!! i learn something about sequence even though i confusing other types of sequence but still i try to appreciate it..thank you so much!!!

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thanks. .I’ve learn lots of sequences. .Like arithmetic and geometric sequences. .I wish many people/pupils can read this letter so that they can learn and understand about sequences.

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thanks for all of your help about sequences

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