Introduction to Sequence and Series

In Mathematics, sequence and series are important concepts, and it has a wide range of applications in the field of Physics, Finance, Statistics. We have studied different types of numbers, such as even numbers, odd numbers, etc. The list of even and odd numbers follows a specific sequence. For instance, the list of even numbers is 2, 4, 6, 8, 10, etc. We can say it as a sequence with the common difference of 2. Here, we are going to discuss the sequence and series and their types.

Sequence 

The sequence is defined as an arrangement of any object or the set of numbers, which follows specific rules. In other words, the sequence is defined as a list of numbers in a particular order. The sequence is also known as “Progression”. The number of elements present in the sequence is called the length of the sequence. The sequence may be finite or infinite. If the difference between two consecutive numbers in the sequence is positive, the sequence follows an ascending order. If the difference between two consecutive numbers is negative, the sequence follows a descending order. 

For example, 1, 3, 5, 7, 9 is a sequence.

Here, the sequence is finite with 5 terms and each term is increased by 2. Hence, the common difference is 2.

Series

The sum of the terms present in the sequence is called a series. The series may also be finite or infinite, depending on the type of sequence. If a sequence is finite, then a series is also a finite series. If the given sequence is infinite, then the series is infinite.

Consider the finite sequence, 4, 8, 12, 16; then the finite series is represented by 4+8+12+16.

For example, if the sequence is infinite, say 1, 2, 3, …., then the infinite series is given by 1+2+3+...

Classification of Sequence and Series 

The four important types of sequences are AP, GP, HP, and Fibonacci number. Now, let us discuss in brief about these different types of sequences.

Arithmetic Sequence:

The arithmetic sequence, also called arithmetic progression is defined as a list of numbers, in which the difference between 2 continuous numbers is constant. For example, 3, 5, 7, 9 is an arithmetic sequence with the common difference of 2. (i.e.) the difference between two consecutive numbers is 2.

Geometric Sequence:

In GP, every element in the list of numbers is determined by multiplying the previous term using the constant value. Here the constant value is called the common ratio. For example, 1, 2, 4, 8, 16 is a geometric sequence, such that every term is obtained by multiplying the previous term by 2.

Harmonic Sequence:

The harmonic sequence is the reciprocal of the arithmetic sequence. In general, if a, b, c, d is an arithmetic sequence, then the harmonic sequence is represented by 1/a, 1/b, 1/c, 1/d.

Fibonacci Numbers:

The Fibonacci numbers, also known as the Fibonacci sequence is a list of numbers in which the next number is found by adding the two leading numbers. For example, the sequence of Fibonacci numbers is given by 5, 8, 13, 21, … and so on.

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