Sequence and series are one of the basic topics in Arithmetic. Here, you will learn the Sequence and Series | Progression | Type of Sequence.

**Sequence**

A Sequence is an arrangement of numbers in a definite order according to some rule.

Example: 5, 5.5, 5.55, 5.555—————etc.

The various numbers that occur in the sequence are known as its **terms**. The terms are denoted by a_{1}, a_{2}, a_{3} — or t_{1}, t_{2}, t_{3} —

The nth term is the number at the nth position of the sequence and as** t _{n}** or

**a**.

_{n}We define a sequence is non-empty set x to be

map f: n-> x or a map f: N-> x

If X = R is called a **real sequence**.

If X = C is called a **complex sequence**.

A sequence that contains the infinite number of terms is known as an **infinite sequence**. In such sequence no terms is uncountable.

**Example:** a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, ———-

A sequence that contains the finite number of terms is known as an **finite sequence**. In such sequence no terms is countable.

**Example:** a_{1}, a_{2}, a_{3} —a_{n}

**Types of Sequence**

Most common examples of sequences are:

1. ** Arithmetic Sequences:** If the difference between two consecutive terms is a constant is called Arithmetic Sequences.

**Example:** 2, 4, 6, 8, 10……….

2. **Geometric Sequences:** A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.

**Example:** 2, 4, 8, 16, 32, 64………..

3. **Harmonic Sequences:** A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.

**Example:** The sequence 1,2,3,4…. is an arithmetic progression, so its reciprocals *1*1, *1*2, *1*3, *1*4….. are a harmonic progression.

4. **Fibonacci Sequence** A sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

**Example:** 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

**Progression **

Those sequences which follow certain patterns are generally referred to as progression.

**Example:** 2, 6, 10, 14 is a progression.

**Series**

Series is the sum of sequences of terms. The series is finite or infinite according to the given sequence is finite or infinite. Series are represented as **sigma**, which indicates that the summation.

For example, “2, 4 6, 8” is a sequence, with terms “2”, “4”, “6”, and “8”; a series S can be,

S = Sum (2, 4, 6, 8) = 20

I hope, this article will help you a lot to understand the **Sequence and Series | Progression | Type of Sequence**. If you still have any doubts and problems with any topic of mathematics you can ask your problem in the Ask Question section. You will get a reply shortly.

Arithmetic Sequences and Series | Arithmetic Mean

Geometric Sequences and Series | Geometric Mean

Harmonic Sequence | Harmonic Mean

Number Series Concept And Tricks | Problems

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