Types of Sequences and Series

Sequences and series are classified into different types. Some of the most commonly used examples of sequences and series are:

• Arithmetic Sequences and Series
• Geometric Sequences and Series
• Harmonic Sequences and Series
• Fibonacci Numbers

Arithmetic Sequence and Series

An Arithmetic sequence is a sequence where each term of the sequence is formed either by adding or subtracting a common term from the preceding number, and the common term is called the common difference. An arithmetic series is referred to as a series developed by using an arithmetic sequence. 

For example,

2, 5, 8, 11, 14,… is an arithmetic sequence with a common difference of 3, and 2 + 5 + 8 + 11 + 14 +… is the corresponding arithmetic series.

Geometric Sequence and Series

A geometric sequence is a sequence where each term of the sequence is formed either by multiplying or dividing a common term with the preceding number, and the common term is called the common ratio.

A geometric series is referred to as a series developed by using a geometric sequence. Depending upon the number of terms in a geometric progression it is classified into two types, namely, finite geometric progression and infinite geometric progression. 

For example,

1, 5, 25, 125, 625,… is a geometric sequence with a common ratio of 5, and 1 + 5 + 25 + 125 + 625 +… is its corresponding geometric series.

Harmonic Sequence and Series

A harmonic sequence is a sequence where each term of the sequence is the reciprocal of the element of an arithmetic sequence. A harmonic series is referred to as a series developed by using a harmonic sequence. 

For example,

2, 5, 8, 11, 14,… is an arithmetic sequence. Now, the harmonic sequence is 1/2, 1/5, 1/8, 1/11, 1/14,… and its corresponding harmonic series is 1/2 + 1/5 + 1/8 + 1/11 + 1/14 +…

Fibonacci Numbers

Fibonacci Numbers are a sequence of numbers where each term of the sequence is formed by adding its preceding two numbers, and the first two terms of the sequence are 0 and 1. 

As the first term, F₀, and the second term, F₁ of the Fibonacci sequence are 0 and 1, the third term will be, F₂ = F₁ + F₀ = 1 + 0 = 1.

Similarly,

• The fourth term, F₃ = F₂ + F₁ = 1 + 1 = 2
• The fifth term, F₄ = F₃ + F₂ =  2 + 1 = 3
• The sixth term, F₅ = F₄ + F₃ = 3 + 2 = 5

Therefore, the (n+1)^th term of the Fibonacci sequence can be expressed as, F_n = F_n-1 + F_n-2. 

The numbers of a Fibonacci sequence are given as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, . . .

Sequences and Series Formulas: In mathematics, sequence and series are the fundamental concepts of arithmetic. A sequence is also referred to as a progression, which is defined as a successive arrangement of numbers in an order according to some specific rules. A series is formed by adding the elements of a sequence.

Let us consider an example to understand the concept of a sequence and series better. 1, 3, 5, 7, 9 is a sequence with five terms, while its corresponding series is 1 + 3 + 5 + 7 + 9, whose value is 25.

This article explores the sequences and series formulas, including arithmetic, geometric, and harmonic series.