Difference between Sequence and Series: Definition, Examples, Sample Questions

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Sequences and series are employed in both mathematics and everyday life to deal with the list of elements. A series is created by a sequence, which is also known as a progression. One of the fundamental concepts of Arithmetic is sequence and series which are employed to recognize patterns. A series is the sum of the components in a sequence, whereas a sequence is the organized arrangement of numbers methodically and according to certain specified principles. For example, if a four-element sequence is 2, 4, 6, 8, the equivalent series will be 2 + 4 + 6+ 8, with the sum or value of the series being 20. Depending on the set of criteria used to generate the sequence and series, there are numerous sorts of sequences and series. Though to some students, sequences and series might seem identical, there is a distinction between the two as in sequence, the order always matters while it is not so in the case of series. Let's take a closer look to better understand the difference between Sequence and Series.

Table of Content

What is a Sequence?
Types of Sequence
What are series?
Types of series
Difference between Sequence and Series
Conclusion
Things to Remember
Sample Questions

What is a Sequence?

Sequence: A sequence is a set of numbers that have been arranged or sorted in a certain order. The terms in a series refer to the numbers in the sequence, and each term is distinguished from the others by a common difference. The end of the sequence is commonly shown by three linked dots, which indicates that the sequence is not broken and that it will continue further.

For example, 3, 6, 9, 12, 15, ... is a series in which each succeeding number is three times bigger than the preceding one. Hence, the common difference is 3.

Types of Geometric Progression

Geometric Progression

Types of Sequence

There are four types of sequences which are:

Arithmetic Sequence
Fibonacci Sequence
Geometric Sequence
Harmonic Sequence

Each of the four sequences is different from the others and it has its own nomenclature. Let's look at each of these four types of sequences in more detail.

Read More: Sequence and Series

Arithmetic Sequence

An Arithmetic sequence is generated by adding or subtracting a specific number from the previous number. The first term is designated by the letter "a," whereas the common difference is denoted by the letter "d".

Finite Sequences are those that have countable elements and do not continue on indefinitely. 2, 4, 6, 8 is an example of a finite arithmetic sequence.
Infinite Arithmetic Sequence- An infinite arithmetic sequence is one in which the terms run on indefinitely. 1,3,5,7,... denotes an infinite arithmetic sequence.

Geometric Sequence

A geometric sequence is one in which each term is formed by multiplying or dividing a defined integer by the number before it. The common ratio is indicated by "r" and the first term of the geometric sequence is denoted by "a."

Check Important Notes for Permutation and Combination

Harmonic Sequence

A harmonic sequence is an arithmetic sequence in which all of the components' reciprocals create an arithmetic sequence that cannot be zero. The initial term of a harmonic progression is 1/a.

Fibonacci Sequence

The Fibonacci Series is a sort of number sequence in which each term is formed by adding its two preceding parts, and the sequence begins with 0 and 1. Fn = Fn-1 + Fn-2 is the Fibonacci sequence, with F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2.

What are Series?

Series:A series is a collection of numbers that is presented as the sum of the numbers in a given order. As a result, every two numbers in a series are separated by the addition sign '+'. The order of the elements in the series does not matter. If a series represents a finite sequence, it is said to be finite, and if it represents an endless sequence, it is said to be infinite.

For example, 2 + 4 + 6 + 8..., is an infinite series, but 3 + 5 + 7 + 9 is a finite series.

Types of Series

There two main types of series are:

Arithmetic Series

An arithmetic series is the sum of a sequence ai, i = 1, 2,....n, where each term is computed by adding or subtracting a constant d from the preceding one. As a result, for i > 1, ai = ai - 1 + d = ai - 2 + d=............... = a1 + d(i-1)

Geometric Series

Geometric series are the total of all the terms in geometric sequences, i.e., if the ratio between each term and the term before it is always constant, the series is said to be geometric.

The five types of series in mathematics are:

Exponent Series,
Harmonic Series,
Geometric Series,
Alternating Series, and
Power Series

The sequence is a collection of numbers arranged in a certain order or according to a set of criteria. The digits of a sequence are added together to form a series. One phrase can appear many times in a sequence. There are two sorts of sequences: infinite terms sequence and finite terms sequence. The sequence and series will next be defined by adding the sequence's terms. In some instances, the sum of infinite terms in a series is also conceivable.

Let us understand this through an example:

1, 3, 5, 7, 9, 11,... is a sequence in which any two values have a common difference of 2, and the sequence continues indefinitely until the upper limit is specified.

These sequences are known as Arithmetic sequences. Now, if we add the numbers in the sequence 1 + 3 + 5 + 7+ 9..., we get a series of these numbers. Arithmetic series is the most common type of series.

Difference between Sequence and Series

There are many distinctions between sequences and series. They are tabulated as below:

Sequence	Series
The grouping of words in a certain order (i.e., related terms following each other) is referred to as a sequence.	The summation of all elements of a sequence is known as series. There are two types of series: finite and infinite series.
The most crucial aspect of a sequence is the order of elements. As a sequence of 5, 6, 7 differs from 7, 6, 5.	The sequence/order of the items in a series is irrelevant. As the series 5 + 6 + 7, is the same as 7 + 6 + 5.
The sequence's components follow a precise pattern.	The total of the sequence's components is the series.
For instance, 1, 2, 4, 6, 8,... n is said to be in a sequence, whereas 1 + 2 + 4 + 6 + 8... n are said to be in a series.	m1 + m2 + m3 + m4 + m5 + m6 +... + mn can be used to describe a finite series.
The general form of a sequence is [pi]n/∞i=1	The general form of a series is Sn = nr = 1 mr.

The video below explains this:

Sequence and Series Detailed Video Explanation:

Conclusion

Arithmetic progression is a series in which the succeeding terms have a common difference, such as 2, 4, 6, 8, and so on. In a geometric progression, on the other hand, each element of the series is a common multiple of the previous word, such as 3, 9, 27, 81, and so on. Similarly, the Fibonacci Series is a well-known infinite sequence in which each term is formed by adding the two preceding words 1, 1, 3, 5, 8, 13, 21, and so on.

Things to Remember

The sequence is defined as a set of integers organised in a certain way. Every number in the sequence is referred to as a term. Five, ten, fifteen, twenty-five............... The three dots at the end of the pattern indicate that the pattern will continue. The first term is 5, second is 10, third is 15, and so on.
A common difference can exist for each word in the sequence, and the pattern will continue with the common difference.
The order of the elements is important in sequences.
The total of the sequence is defined as the series, where the order of the items is not important.
It signifies the series is specified as a list of numbers separated by an addition symbol.
The series can be categorized as either a finite or infinite series, depending on whether the sequence is finite or infinite.

Read Also: Sequence and Series

Sample Questions

Ques: If the 5th term is 12 and the 7th term is 24, what is the sequence's 6th number? (3 marks)

Ans: The 6th number will be the Arithmetic mean of the two supplied numbers, as the two numbers are given.

AM = 12 + 24 / 2

= 36/2

= 18

As a result, the sixth term will be 18.

Ques: Calculate the geometric mean of the numbers 2 and 18.(3 marks)

Ans: The geometric mean is calculated using a formula.

p = 2 and q = 18

GM = √pq

= √2×18

= √36

= 6

Ques: Highlight the difference between a sequence and a series? (3 marks)

Ans: There is a lot of confusion about the difference between sequence and series, yet it's easy to tell the difference:

A sequence is a format of components in a certain order, whereas a series is the sum of the sequence's constituents.
The order of the items in a sequence is set, whereas the order of the elements in a series is not.
A sequence is represented by the numbers 1,2,3,4,....n, whereas a series is represented by the numbers 1+2+3+4+.....n.
The order of the items in a sequence must be preserved, but the order of the elements in a series is unimportant.

Ques: What is the definition of an arithmetic series ? (3 marks)

Ans: When you sum up all of the terms in a sequence, you obtain an arithmetic series. The resultant values are referred to as the "sum" or "summation." The Latin capital letter "S'' or the Greek letter "sigma" that corresponds to the capital "S'' are used to signify series.

We would write Σ10n=1an to represent the sum of the tenth terms of a sequence an.

Where "n = 1" denotes the "lower index," which indicates that the series begins at 1 and

"n = 10" denotes the "upper limit," which indicates that the last term will be 10.

The term "an" refers to the terms we'll be adding.

It's written as "the total from n equals one to ten".

Ques: In the given sequence, write down the next three terms: 1, 4, 7,....(2 marks)

Ans: 10, 13, and 16 are the following three terms in the series. The difference between 1 and 4 is 3, while the difference between 4 and 7 is 3.

As a result, the sequence's common difference is 3.

Hence, 7+3 equals 10, 10+3 equals 13, and 13+3 equals 16.

Ques: What are the many forms of mathematical series ? (2 marks)

Ans: Mathematical series include :

Arithmetic Series,
Harmonic Series, and
Geometric series,
as well as P-series, exponential, and other forms.

Ques: What is a finite sequence?(1 mark)

Ans: A finite sequence is one in which the number of words in the sequence is finite or fixed.

For instance, 1, 2, 3, 4 is a finite sequence with 4 terms.

Ques: We have a given sequence, S = 1, 3, 5, 7,... Find the twenty-first term. (2 marks)

Ans: The common difference, denoted by the letter d, is two.

The formula for finding the nth term is Tn = a + (n-1)

= 1 + (21-1)2

= 1 + 40

= 41

Ques: Eight and ten are the mathematical and geometric progressions of two integers. Find the figures. (5 marks)

Ans: A.M = 8

G.M = 10

Let the numbers be a and b

The arithmetic is determined by the formula = (a + b)/2

The formula that determines geometric progression= √ab

As per the given values, (a + b)/2 = 8

a + b = 16 --(1)

√ab = 10

ab = 100

Now that we've figured out the formula, (a-b)2 = (a+b)2 - 4ab

= 256 - 400

= - 144

a - b = -12 ---(2)

From 1 and 2 we get a = 2 and b = 14

Ques: Why don't the elements of a series appear in a specific order? (2 marks)

Ans: As the constituents of a series are just additions of the numbers in a sequence, they do not follow any specific order. The addition of two integers usually does not occur in any particular sequence.

Ques: Who was the first to coin the terms "sequence" and "series" ? (1 mark)

Ans: Carl Friedrich Gauss, a mathematician, was the originator of sequence and series.

Related Links:

Sum of n Terms

Geometric Mean

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