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A list of numbers presented in a certain order is called a sequence.
The terms of the sequence are the numbers that are included in the list.
The accumulation of all of the terms in a sequence results in a series.
Sequence and series are both analogous to sets.
However, the key distinction between the two is that in a sequence, individual terms may recur several times but in a different order each time.
The number of words, which may be either finite or infinite, determines the length of a sequence, which is equal to that number.
Explain what are sequence and series
Sequences can be of any length. In other terms, we can say that a sequence is a collection of things or items that have been ordered in a manner that is both systematic and sequential. Eg: a¹,a²,a³, a4…….
Having said that, however, The term “series” denotes the aggregate of all the elements that are at our disposal, or we can say that a series is synonymous with the aggregate of all the elements that are at our disposal in the sequence.
The arithmetic progression is an example of both a sequence and a series, and it is among the most common of both instances.
Eg:
If something like a¹,a²,a³, a4 etc. is regarded to be a sequence, then the total of all the terms in that sequence, a¹ + a² + a³ + a4, etc. is considered to be a series.
The key distinctions between Sequence and Series
The terms “sequence” and “series” are sometimes used interchangeably, however, the difference between the two can be understood quite clearly by considering the following:
The difference between a sequence and a series is that a sequence is a certain format of items in some definite order, whereas a series is the sum of the parts that make up a sequence.
In a sequence, the order of the elements is always the same, whereas, in a series, the order of the elements can change at any time.
A sequence is denoted by the notation 1, 2, 3, 4,… n, whereas a series is denoted by the notation 1 + 2 + 3 + 4 +…. n.
In a sequence, it is necessary to keep the elements in the correct order; but, in a series, the order of the elements is not as significant.
The following considerations are important in developing an in-depth comprehension of the ideas of sequence and series.
an is considered to be the first term in an arithmetic sequence or series, d is considered to be a common difference, an is considered to be the nth term, and n is considered to be the total number of terms.
The arithmetic sequence can be represented in its most basic form as a, a+d, a+2d, a+3d, etc.
In a geometric progression, obtaining each consecutive term is accomplished by multiplying the common ratio by the phrase that came before it.
The formula for the nth term of a geometric progression, whose initial term is a and whose common ratio is r, is a = arn, where a and r are the two terms that come before it.
The infinite GP formula has a sum that may be expressed as An = a/(1r), provided that |r| is less than 1.
A Succession of Numerical Values
In the field of mathematics, an ordered set of numbers is referred to as a sequence or progression.
Each individual number that is included in the series is referred to as the nth term, where n represents the position of that particular term in the sequence.
Different kinds of successions
Mathematically speaking, one obtains the next term in the sequence by adding a constant to the term that came before it.
The difference is the name given to this unchanging factor.
The next phase in the series can be derived by multiplying the previous term by a constant.
This is a geometric concept. The name given to this constant is the ratio.
The definition of the series is derived using recursion, making it recursive.
Examples
Both the even number sequences (0, 2, 4, 6, 8,…) and the odd number sequences (1, 3, 5, 7,…) are mathematically equivalent, with the exception that the even number sequences differ by a factor of 2.
The geometric progression of the powers of two (2, 4, 8, 16, 32, etc.) is defined by the ratio r = 2, and it begins with 2.
The series 2, 1, 0.5, 0.25, 0.125, 0.0625… has a geometric ratio of r = 0.5, making it a geometric sequence.
There is a cycle of repetition in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21,…).
Every word is obtained by adding the two terms that came before it, with the exception of the initial two terms, which are always 1 by definition.
Finite and Infinite Sequence
If a sequence does not have an inherent limit on the number of terms it can contain, then it is said to be infinite.
Finite sequence: {4,8,12,16,…,64}
The number 4 comes first in the sequence, while the number 64 comes last. A sequence is said to be finite if it can be reduced to a single final term.
Infinite sequence: {4,8,12,16,20,24,…}
The number 4 is the initial number of the sequence.
The sequence does not have a concluding phrase because it ends with a “…”, which suggests that it continues indefinitely. It is a sequence that goes on forever.
Conclusion
A set of numbers that are arranged in a predetermined order is an example of what is referred to as a sequence.
It is regarded to be a collection of components that organises itself according to a particular structure.
On the other hand, one way to define a series is as the total number of elements that make up a certain sequence.
The order in which elements are presented in a sequence is critical to the formation of the sequence.
Despite the fact that the sequence of the components that make up a series is not particularly significant.
