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When it comes to mathematics, a sequence is a list of objects (or events) that have been sorted in a sequential manner, with each item appearing either before or after every other member. To put it another way, in mathematical terms, a sequence is a function having a domain equal to the set of positive integers.
Sequence and series: A brief note
A series is a collection of terms that are arranged in a particular order. To put it another way, a series is a list of numbers with additional operations performed on them in between.
A sequence is a list of integers that is arranged in a specific order. The terms of the series are represented by the numbers in the list. The terms in a series are typically referred to as an, with the index denoted by the subscripted letter I or n. As an example, the second term of a series might be designated as a2, while the twelfth term would be designated as a12.
Various types of sequences and series
Although there are many other sorts of sequences and series, we will focus on some of the most special and frequently used sequences and series in this part. The following are examples of sequences and series:
1. Arithmetic Sequences and Series
2. Geometric Sequences and Series
3. Harmonic Sequences and Series
Arithmetic Sequence and Series
As defined by the Common Difference definition, an arithmetic sequence is a sequence in which the consecutive terms are either additions or subtractions of the common term known as the common difference. For example, the numbers 1, 4, 7, 10, and so on are all arithmetic sequences. The term “arithmetic series” refers to a series that is constructed by utilising an arithmetic sequence. For example, the series 1 + 4 + 7 + 10… is an arithmetic series.
Geometric Sequences and Series
A geometric sequence is a sequence in which the items that follow each other have a common ratio. For example, the arithmetic sequence 1, 4, 16, 64 is an arithmetic sequence. A geometric series is a series that is constructed by employing a geometric sequence. For example, the series 1 + 4 + 16 + 64… is a geometric series, and so on. Finite geometric progression and infinite geometric series are the two types of geometric progression that can be applied.
Harmonic Sequence and Series
A harmonic sequence is a sequence that is generated by taking the reciprocal of each term of an arithmetic sequence and rearranging the terms in the sequence. A harmonic sequence is composed of numbers such as 1, 1/4, 1/7, 1/10, and so on. A harmonic series is a series that is constructed by employing harmonic sequences. For example, a harmonic series is 1 + 1/4 + 1/7 + 1/10….
Conclusion
Sequence: For example, consider the following: A finite sequence is a list of numbers that terminates at the conclusion of the list: 1, 2, 3, 4, 5, 6…10. An infinite sequence, on the other hand, is never-ending, as in a1, a2, a3, a4, a5, a6……an…..
Series: The term a1 + a2 + a3 + a4 + a5 + a6 +……and represents a finite number of terms in a finite series, which is expressed as a1, a2, a3,…an. It is not possible to have a finite number of elements in an infinite series, for example, a1 + a2 + a3 + a4 + a5 + a6 +……an +…..
