Types of Sequences

In types of sequences, the word, “ sequence ” is used in much the same way as it is in ordinary English. When we say that a collection of objects is listed in a sequence, we usually mean that the collection is ordered in such a way that it has an identified first member, second member, and third member and so on.

For example: Population of human beings at different times form a sequence.

Here we are going to learn about various types of sequences.

The number of person’s ancestors for the first, second, third… Tenth generation’s are 2, 4, 8, 16, 32 …1024. These numbers form what we call a sequence.

Thus, the terms of the sequence of person’s ancestors mentioned above are:

a1 = 2, a2 = 4, a3 = 8… a10 = 1024.

Here three types of sequences are there,

1. Finite sequence,
2. Infinite sequence,
3. Fibonacci sequence

Finite sequence: A type of sequence containing finite number of terms is called a finite sequence.

For example: Sequence of ancestors is a finite sequence since it contains 10 terms.

(a fixed number).

Infinite sequence: A type of sequence, which is not a finite sequence, means it is called infinite sequence.

For example: The sequence of successive quotients mentioned above is an infinite sequence, infinite in the sense that it never ends.

Often, it is possible to express the rule, which yields the various terms of a sequence in terms of algebraic formula. Consider for instance, the sequence of even natural numbers 2, 4, 6 …

Here,

a1 = 2 = 2 × 1, a2 = 4 = 2 × 2, a3 = 6 = 2 × 3, a4 = 8 = 2 × 4 … a24 = 48 = 2 × 24, and so on.

Fibonacci sequence:

In above example, we see that the nth term of this sequence can be written as an = 2n, where n is a natural number. Similarly, in the sequence of odd natural numbers 1, 3, 5… the nth term is given by the formula, an = 2n – 1, where n is a natural number.

In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. Has no visible pattern, but the sequence is generated by the recurrence relation given by

a1 = a2 = 1

a3 = a1 + a2

an = an – 2 + an – 1, n > 2

This type of sequence is called Fibonacci sequence.

This is one way of categorizing sequence based on whether the terms are finite or infinite.

There are other ways of categorizing depending on how the terms are obtained or related. The main types are:

Arithmetic Sequence : This is a type of sequence where any term except the first term is obtained by adding a fixed number to the previous term. Example : 4, 7,10....

Geometric Sequence : This is a type of sequence where any term except the first term is obtained by multiplying a fixed number to the previous term. Example : 4, 12,36....

Harmonic Sequence : This is a type of sequence where the reciprocal of the terms from an arithmetic sequence.