Types of Sequence and Series

One of the foundational tenets of arithmetic is the concept of sequence and series. 

One of the most typical illustrations of a sequence and a series is an arithmetic progression.

A list of elements or objects that have been arranged in a sequential manner is an example of what is meant by the term “sequence.”

A series is said to be geometric if there is a relationship between the ratio of each pair of consecutive terms in the series and a constant function of the summation index.

 The ratio is a rational function of the summation index in the more general case of the expression that generates what is known as a hypergeometric series out of all the terms that are arranged in a sequence. 

However, there must be a clear connection between each of the elements of the sequence in order for it to be valid.

The difference between Sequence and Series

A group of numbers or other items that are arranged in a specific order and are then followed by a set of guidelines constitutes a sequence.

If the terms in a series are indicated by a1, a2, a3, a4, etc., then the location of the term is indicated by 1, 2, 3, 4, etc.

One way to define a series is according to the number of terms it contains, which results in either a finite or an endless sequence.

If a1, a2, a3, a4, and so on is a sequence, then the series that corresponds to it is provided by

SN = a1+a2+a3 + .. + aN

Note that the nature of the sequence determines whether the series is finite or infinite. 

Different kinds of sequential and serial order

There are many distinct kinds of sequences and series. Namely:-

• Sequences and Series of Arithmetic Operations

• Sequences and series of geometric objects

• Sequences and Series of Harmonic Objects

Sequence-based on Arithmetic

The arithmetic sequence is a sequence in which the common difference between any two successive terms remains the same.

A sequence is a collection of numbers that are arranged in a specific order. 

For instance, the number sequence 1, 6, 11, 16,… is an example of an arithmetic sequence since there is a pattern in which each number is obtained by adding 5 to the term that came before it. 

There are two different formulas for arithmetic sequences.

A mathematical expression that can be used to locate the nth term of a sequence.

A mathematical expression that can be used to find the sum of an arithmetic sequence’s first n terms

An example of an Arithmetic Sequence

Consider the sequence 3, 6, 9, 12, 15,…. 

This is an example of an arithmetic sequence since each phrase in the sequence can be created by adding a constant number (3) to the value of the preceding item in the sequence.

Here,

The initial word, which reads a = 3,

The standard deviation, d, is calculated as follows: (6 – 3) = (9 – 6) = (12 – 9) = (15 – 12) =… = 3

As a result, the notation for an arithmetic sequence looks like this: a, a + d, a + 2d, a + 3d, etc.

The Geometric Sequence

A geometric sequence can be added up to form a geometric series, which is the sum of any number of terms, finite or infinite. 

The appropriate geometric series for the sequence of geometric constants a, ar, ar2,…, arn-1,… is the sequence a + ar + ar2 +…, arn-1 +…. 

We are aware that “sum” might also be interpreted as “series.” 

In particular, the phrase “geometric series” refers to the addition of all the terms that share a ratio with every other pair that is immediately adjacent to them. 

There is the possibility of an infinite and a finite number of steps in a geometric series. 

The Sequence and Series of Harmonics

A series is said to be harmonic if it is generated by taking the reciprocal of each term of an arithmetic sequence, as this results in the formation of the sequence. 

One example of a harmonic sequence includes the numbers 1, 1/4, 1/7, 1/10, etc. 

A series that is created by the use of a harmonic sequence is referred to as a harmonic series.

 For instance, the sequence 1 + 1/4 + 1/7 + 1/10… is an example of a harmonic series.

Some Examples of Sequence and Series

If the numbers 4, 7, 10, 13, 16, 19, 22,…… are in order, then find:

Difference between the nth and 21st terms common nth term

Solution: Given sequence is, 4,7,10,13,16,19,22……

a) The most common difference is 7, which minus 4, equals 3.

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by the equation Tn = a + (n-1)d, where “a” is the first term and “d” is a common difference.

 c) The nth term of the arithmetic sequence is denoted by the term Tn and is given by the equation

 Tn = a + (n-1)

Tn = 4 + (n – 1)

21st term as: T21 = 4 + 3n – 3 = 3n + 1 

4+60 = 64.

Fibonacci Sequence

Leonardo Pisano Bogollo, an Italian mathematician, was the first person to discover the Fibonacci sequence (Fibonacci).

 The Fibonacci sequence is a sequence of whole numbers that starts with 0 and continues on to 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. 

The name given to this never-ending sequence is the Fibonacci sequence. 

In this situation, each word is the sum of the two terms that came before it, beginning with 0 and 1.

 This phenomenon is often referred to as “nature’s secret code.”

Conclusion

A group of numbers that are arranged in a specific order according to a predetermined set of guidelines is referred to as a sequence. 

When terms from a sequence are added together, a series is created.

 One and the same term can appear multiple times at various points in the sequence.

 Sequences can be either infinite or finite, and their terms are added together to form a series. 

Sequences can be either infinite or finite, and their terms are used to define the series. 

In some circumstances, it is also feasible for a series to have a sum of infinite terms.