Sequence and series are utilized in both mathematics and everyday life. A sequence is also known as progression, and by sequence, a series is created. Sequence and series are fundamental notions in mathematics. Sequences are the grouped arrangement of numbers in accordance with particular criteria, whereas a series is the sum of the sequence’s constituents. For instance, 2, 4, 6, 8 is a sequence of four elements, and the corresponding series is 2 + 4 + 6 + 8, with a sum or value of 20.
There are different types of sequences and series based on the set of rules employed to create them. Detailed explanations of sequence and series follow.
What Are Sequence and Series?
The sequence is the group or arrangement of numbers in a specific order or according to a set of criteria. Sequence terms are added together to make a series. An individual phrase can appear multiple times in a series. There are two sorts of sequences, namely infinite sequences and finite sequences, and series are defined by adding the terms of a sequence. In some instances, the sum of infinite terms in a series is also conceivable.
Allow me to illustrate with an example. There is a common difference of 2 between any two terms in the series 1, 3, 5, 7, 9, 11,…, and the sequence increases to infinity until an upper limit is specified. These sequences are referred to as arithmetic sequences. Now, if we add the numbers in the sequence, such as 1 + 3 + 5 + 7 + 9, we will have a series of these numbers. These series are referred to as arithmetic series. Several instances of sequence and series are illustrated in the graphic below:
Difference Between Sequence and Series
The important differences between sequence and series are explained in the table given below:
| Sequence | Series |
| In sequence, elements are placed in a particular order following a particular set of rules. | In series, the order of the elements is not necessary. |
| It is just a collection (set) of elements that follow a pattern. | It is a sum of elements that follow a pattern. |
| The order of appearance of the numbers is important. | The order of appearance is not important. |
| Example: Harmonic sequence: 1, 1/2, 1/3, 1/4, … | Example: Harmonic series: 1 + 1/2 + 1/3 + 1/4 + … |
Types of Sequence and Series
There are various types of sequences and series, in this section, we will discuss some special and most commonly used sequences and series. The types of sequence and series are:
- Arithmetic Sequences and Series
- Geometric Sequences and Series
- Harmonic Sequences and Series
- Fibonacci Numbers
Arithmetic Sequence and Series
An arithmetic sequence is a sequence in which successive terms are either the addition or subtraction of a common term called the common difference. For instance, 1, 4, 7, 10,… is an example of an arithmetic sequence. A series produced using an arithmetic sequence is known as an arithmetic series, such as 1 + 4 + 7 + 10…
Geometric Sequence and Series
A geometric sequence is a sequence in which each succeeding phrase has the same ratio. Sequences 1, 4, 16, 64,… is an example of arithmetic sequences. A series built using a geometric sequence is known as a geometric series, such as 1 + 4 + 16 + 64… There are two types of geometric progression: finite geometric progression and infinite geometric series.
Harmonic Sequence and Series
A harmonic sequence is a series in which each term of an arithmetic sequence is replaced by its reciprocal. The series 1, 1/4, 1/7, 1/10, etc. is an example of a harmonic sequence. A series built using a harmonic sequence, such as 1 + 1/4 + 1/7 + 1/10…, is known as a harmonic series.
Fibonacci Numbers
Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2
Sequence and Series Formulas
There are numerous formulas associated with diverse sequences and series. Using these, we can determine a set of unknown variables, such as the first term, the nth term, and common parameters, among others. Different formulas apply to each type of sequence and series. Formulas associated with diverse sequences and series are described below:
Arithmetic Sequence and Series Formula
The various formulas used in arithmetic sequence are given below:
| Arithmetic sequence | a, a + d, a + 2d, a + 3d, … |
| Arithmetic series | a + (a + d) + (a + 2d) + (a + 3d) + … |
| First term: | a |
| Common difference(d): | Successive term – Preceding term or an – an-1 |
| the nth term, an | a + (n-1)d |
| Sum of arithmetic series, Sn | \( \frac{n}{2}\left[2a+\left(n-1\right)d\right]\) |
Geometric Sequence and Series Formulas
The various formulas used in geometric sequence are given below:
| Geometric sequence | a, ar, ar2,….,ar(n-1),… |
| Geometric series | a + ar + ar2 + …+ ar(n-1)+ … |
| First term | a |
| Common ratio | r |
| nth term | ar(n-1) |
| Sum of geometric series | Finite series: Sn = a(1−rn)/(1−r) for r≠1, and Sn = an for r = 1Infinite series: Sn = a/(1−r) for |r| < 1, and not defined for |r| > 1 |
Sequence and Series Tips
The following points will aid in your comprehension of the ideas of sequence and series.
- In an arithmetic sequence and series, a represents the first term, d represents a common difference, n represents the number of terms, and a represents the nth term.
- The general representation of the arithmetic sequence is a, a+d, a+2d, a+3d,…
- In a geometric progression, each new term is obtained by multiplying the common ratio by its predecessor.
- Each successive term is obtained in a geometric progression by multiplying the common ratio by its preceding term.
- The formula for the nth term of a geometric progression whose first term is a and common ratio is r is \( a_n=ar^{n-1}\)
- The sum of the infinite GP formula is given as \( s_n=\frac{a}{1-r}\) where |r|<1.
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Examples of Sequence and Series :
Example 1: What will be the 15th term of the arithmetic sequence -3, -(1/2), 2…. using sequence and series formula?
Solution: Given a = -3, d = -(1/2) -(-3) = 5/2, n = 15
Using the formula for the nth term of an arithmetic sequence:
\( a_{n} = a+(n-1)d\)
Putting the known values:
\( a_{15} = -3 +(15-1) 5/2\)
\( a_{15}= 32\)
Answer: The 15th term of the given arithmetic sequence is 32.
Example 2: Find the next term of the given geometric sequence: 1, 1/2, 1/4, 1/8 … using sequence and series formula
Solution: Given: a = 1, r = (1/2)/1 = 1/2
To find: 5th term
Using the formula for the nth term of a geometric sequence and series:
\( a_n=ar^{n-1}\)
Putting the known values in the formula:
\( a_{5} = 1(1/2)(5-1)\)
\( a_{5}= (1/2)(4)\)
\( a_{5}= 1/16\)
Answer: The next term of the sequence is 1/16.
Example 3: Find the sum of the infinite geometric series -1 + 1/2 – 1/4 + 1/8 – 1/16 + …
Solution:
The common ratio of the given series is, r = -1/2.
Here, |r| = |-1/2| = 1/2 < 1.
Using the sequence and series formulas,
Sum of the given series = a / (1 – r)
= -1 / (1 – (-1/2))
= -1 / (3/2)
= -2/3
Answer: -2/3
FAQ ( Frequently Asked Questions ) on Sequence and Series
Q.1 What are Sequence and Series?
Sequence and series are utilized in both mathematics and everyday life. The sequence is the group or arrangement of numbers in a specific order or according to a set of criteria. Sequence terms are added together to make a series.
Q.2 What are Some of the Common Types of Sequences?
A few popular sequences in maths are:
Arithmetic Sequences
Geometric Sequences
Harmonic Sequences
Fibonacci Numbers
Q.3 What are Finite and Infinite Sequences and Series?
Sequences: A finite sequence is a sequence that contains the last term such as a1, a2, a3, a4, a5, a6……an. On the other hand, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..
Series: In a finite series, a finite number of terms are written like a1 + a2 + a3 + a4 + a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +…..
Q.4 How to represent the arithmetic sequence?
If “a” is the first term and “d” is the common difference of an arithmetic sequence, then it is represented by a, a+d, a+2d, a+3d, …
Q.5 What is the Difference Between Sequence and Series?
In sequence, items are arranged in a specific order according to a specific set of rules; a specific pattern of the numbers is significant, as is the order in which the numbers appear. In a series, the order of the elements is not crucial, and neither is the pattern of the numbers nor the order of appearance. Example of sequence: Harmonic sequence: 1, 1/2, 1/3, 1/4, 1/5, 1/6… . Example of series: Fourier series: f(x) = 4h/π ( sin(x) + sin(3x)/3 + sin(5x)/5 + … )
Q.6 What is Geometric Sequence and Series?
A geometric sequence is a sequence in which each succeeding phrase has the same ratio. The sequence 1, 4, 16, 64,… is an example of an arithmetic sequence. A series built using a geometric sequence is known as a geometric series, such as 1 + 4 + 16 + 64… There are two types of geometric progression: finite geometric progression and infinite geometric progression.
