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One of the most essential notions in Arithmetic is the concept of sequence and series. Sequences are grouped arrangements of numbers that are done in an orderly manner according to some specified criteria, whereas a series is the sum of the components in a sequence. The following are some examples of four-element sequences: 2, 4, 6, 8, where 2 + 4 + 6+ 8 is a four-element series with a sum of the series or a value of the series equal to twenty. There are many different types of sequences and series, each of which is distinguished by the set of rules employed to construct it. Below, the concepts sequence and series are described in further depth.
Types of Sequences
1. Arithmetic Sequence
It is a sequence in which every term is found by adding or subtracting a definite number(common difference) from the preceding number.
x1, x1+ m , x1+ 2m, x1+ 3m,….. x1+ (n-1)m and so on, where m is the common difference.
For example- 1, 4, 7, 10,13,16…….
Here, we observe that the common difference is 3.
2. Geometric Sequence
A sequence in which each of the terms in a series is obtained by either multiplying or dividing a constant number with the former one is said to be a geometric sequence. The common ratio of a GP is obtained by taking the ratio between any one term in the sequence and dividing it by the previous term.
For example- 2, 4, 8, 16,32 …
Here, the common ratio or factor is 2.
General form- a, am, am2, am3, am4,…, amn
Where, a is the first term m is the common ratio between the terms.
3. Harmonic Sequences
A harmonic sequence is a sequence where the reciprocals of all the terms of the sequence form an arithmetic sequence.
4. Fibonacci Number Sequence
Fibonacci numbers form a sequence of numbers in which each of the terms is obtained by adding two preceding terms. The sequence starts with 0 and 1.
For example-
F1 = 0,
F2 = 1,
F3 = 1,
F4 = 2,
…
Fn = Fn-1 + Fn-2
Series
Series is defined as the sum of all the elements in a sequence or it is defined as the sum of the sequence where the order of the terms does not matter. The series can be classified as finite or infinite depending on the types of sequence made of, whether it is finite or infinite.
For example, 1+3+5+7+… is a series. The different types of series are:
Geometric series
Harmonic series
Power series
Alternating series
Exponent series (P-series)
What is the difference between sequence and series?
A sequence is defined as the grouping or sequential arrangement of numbers in a specified order or according to a set of criteria. The terms of a sequence are joined together to form a series. A single phrase can appear in many places within a sequence. A finite sequence is different from an endless sequence. By merging the terms of a sequence, a series is formed. It is also possible to have a series with an infinite number of terms under particular circumstances.
To further understand this, consider the following scenario. The sequence 1, 3, 5, 7, 9, 11,… is generated when any two words have a common difference of two. The sequence will continue to expand indefinitely unless an upper limit is given. Arithmetic sequences are a type of sequence that represents numbers. After that, we can combine the numbers in the sequence, such as 1+3+5+7+9…, to form a series of the numbers in the sequence. The sorts of series that come within this category are arithmetic series.
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. |
As an illustration, consider the following harmonic sequence: 1, 1/2, 1/3, 1/4,… | As an illustration, consider the harmonic series: 1 + 1/2 + 1/3 + 1/4 +… |
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 +……an represents a finite number of terms in a finite series, which is expressed as a1, a2, a3, a4, a5, a6……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 +…..
