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The concept of sequence and series is one of the most fundamental concepts in Arithmetic. A sequence is an itemised collection of components in which repetitions of any kind are permitted, whereas a series is the sum of all of the elements. Sequence and series are commonly used to describe arithmetic progression, which is one of the most common examples. Briefly stated, a sequence is a series of elements or objects that have been organised in chronological order. A series can be thought of as the sum of all the terms in a sequence, which is a strong generalisation. It is necessary, however, for all the terms in the sequence to have a clear link with one another.
Sequences and Series
Generally speaking, a sequence is an arrangement of items or a collection of numbers in a specific order that is followed by a rule. If the terms of a sequence are denoted by the letters a₁, a₂, a₃, a₄,…etc., then the numbers 1, 2, 3, 4,…. denote the position of the term in the sequence.
A series can be classified as either a finite sequence or an infinite sequence depending on the number of terms in it.
The series corresponding to the sequences a₁, a₂, a₃, a₄, and so on is given by the formula
Sₙ = a₁+a₂+a₃ + .. + aₙ
Depending on whether the sequence is finite or infinite, the series is either finite or infinite.
Types of Sequences and Series
Although there are many other 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:
- Arithmetic Sequences
- Geometric Sequences
- Harmonic Sequences
- Fibonacci Numbers
Arithmetic Sequences
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 formed by using an arithmetic sequence. For example, the series 1 + 4 + 7 + 10… is an arithmetic series.
Geometric Sequences
A geometric sequence is a sequence in which the items that follow each other have a common ratio. For example, the numbers 1, 4, 16, 64, and so on are all arithmetic sequences. A geometric series is a series that is formed by using 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 Sequences
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, ¼, 1/7, 1/10, and so on. A harmonic series is a series that is formed by using harmonic sequences. For example, 1 + ¼ + 1/7 + 1/10… is a harmonic series.
Fibonacci Numbers
It is important to note that the Fibonacci series is a sequence of integers in which each element is obtained by adding two elements that came before it, and the sequence begins with the numbers 0 and 1. Sequence is defined as Fₙ = Fₙ₋₁ + Fₙ₋₂, with F₀ = 0 and F₁ = 1.
Difference Between Sequence and Series
These are the most significant distinctions between sequence and series, as seen in the table below:
Sequence | Series |
During the process of sequencing, items are placed in a specific order according to a specific set of rules. | It is not necessary to arrange the elements in a sequential manner in a series. |
It is simply a collection (set) of items that are organised in a certain way. | It is made up of a collection of pieces that follow a pattern. |
The order in which the numerals occur is very crucial. | It makes no difference in whatever order the items appear. |
Example: Harmonic sequence: 1, ½, 1/3, ¼, … | Example: Harmonic series: 1 + ½ + 1/3 + ¼ + … |
Points to Remember
Considering the following topics will assist you in better understanding the concepts of sequence and series.
- In an arithmetic sequence and series, an is represented as the first term, d is represented as a common difference, aₙ is represented as the nth term, and n is represented as the total number of terms in the series
- For the most part, the arithmetic sequence can be represented by the letters a, a+d, a+2d, a+3d…, etc
- Each consecutive term in a geometric progression is derived by multiplying the common ratio of its preceding term by the number of terms in the progression
- The formula for the nth term of a geometric progression whose first term is a and common ratio is r is a = arn1 where an is the first term and r is the common ratio
- A simplified version of the infinite GP formula is given as Sn = a/(1r), where |r| < 1
Conclusion
When it comes to actual life, the arithmetic sequence is significant because it allows us to make sense of things by looking for patterns in them. For example, arithmetic sequences are a fantastic foundation for defining many different things like time which has a standard difference of one hour. When replicating systematic events, it is also crucial to use an arithmetic sequence.
