Sequences and Series: Types & Formulas of Arithmetic, Geometric & Harmonic with Solved Examples
Saurav Sinha
| Updated: Dec 2, 2022 17:06 IST
Sequence and series contribute a major part of mathematics, where sequence is the arrangement of objects or items in a progressive manner and series is the sum of all the terms in the particular sequence.
What are Sequence and Series?
Sequence and series are employed in basic to higher-level mathematical concepts. A sequence is also recognized as a progression while a series is generated by the sequence.
Sequence: A sequence is an organization of any objects/elements/set of digits in a particular order accompanied by some rule.
Example of sequence, if \(a_1,\ a_2,\ a_3,\ a_4,\dots\dots\dots\) etc indicate the terms of a sequence, then . \(1,\ 2,\ 3,\ 4,\dots\) signifies the position of the term in the sequence.
Finite or Infinite: When the sequence continues with endless terms then it is named an infinite sequence, otherwise, it is a finite sequence.
Finite or Infinite Sequence Examples:
- {1, 2, 3, 4, 5, 6 …} is a simple example of an infinite sequence.
- Whereas {1, 3, 5, 7, 9, 11} is an example of a finite sequence.
- Likewise {a, b, c, d, e, f, g, h} is an example of an alphabetic sequence.
Series: Sequences and series are counted under some of the basic concepts in arithmetic. Continuing the definition of sequence if \(a_1,\ a_2,\ a_3,\ a_4,\dots\dots\dots\) is a sequence, then the analogous series is given by \(a_1+a_2+a_3+….\).
The series is either finite or infinite depending on whether the sequence is finite or infinite.
Types of Sequence and Series
A Sequence is like a set, but the terms are in order and the identical value can appear multiple times. Sequences also apply the same notation as that of sets: the list of the elements is separated by a comma and placed around curly brackets e.g {2, 4, 6, …}. The curly brackets { } are sometimes also known as set brackets or braces. A Sequence usually has a rule, which is a way to determine the value of each term. For example, the sequence {2, 4, 6, 8, …} starts at 2 and skips 2 every time. Let us start with the types of series and types of sequences.
There are 3 types of sequences and series:
- Arithmetic Sequence and Series
- Geometric Sequence and Series
- Harmonic Sequence and Series
Let’s look at each of them one by one to have a detailed analysis, starting with arithmetic sequences and series.
Arithmetic Sequence and Series
The first type is arithmetic sequence and series.
Arithmetic Sequence: A sequence is termed an arithmetic sequence if the difference between the term and its previous term is always the same. In general, we can address an arithmetic sequence as:
\(\left\{a,\ a+d,\ a+2d,\ a+3d,\dots\right\}\)
where “a” is the first term of the sequence, and “d” is the difference or the common difference between the terms. In an arithmetic sequence, the relation between the first term, common difference and the nth term is:
\(x_n=a+(n-1)\ d\)
Example of Arithmetic Sequence: 4, 8, 12, 16, 20, 24, 28……..
In the above example, the difference between the successive terms is 4.
Arithmetic Series: In an arithmetic sequence, if there is the summation of the given terms then it is called an arithmetic series. In simple terms, we can say that an arithmetic series is an aggregate of a sequence \(a_i\), where i = 1, 2,….n.
Here, each term is calculated from the earlier one by adding or subtracting a constant number denoted by d.
If an arithmetic sequence is taken as: a, a + d, a + 2d, a + 3d, …
Then the arithmetic series is understood as:
a + (a + d) + (a + 2d) + (a + 3d) + …
Geometric Sequence and Series
In geometric sequence and series, we will learn about the second type of classification which is geometric sequence and series. Starting with the geometric sequence definition.
Geometric Sequence: A sequence where every successive term possesses a fixed ratio between them is called a geometric sequence. In other words, a sequence where every term can be obtained by multiplying or dividing a particular number with the preceding number is called a geometric sequence.
The first term of the geometric sequence is termed as “a”, and the common ratio is denoted by “r”. In general, we can address a geometric sequence as:
\(a,\ ar,\ ar^2,\ ar^3,\dots.\ ar^{n-1}\) and: \(a_n=ar^{n-1}.\)
where “a” is the first term, and “r” is the factor between the elements termed the common ratio.
Example of Geometric Sequence: 3, 6, 12, 24, 48.
In the above example, the difference is a common ratio between each term is 2, and the first term here is 3. Finally, the formula becomes \(a_n=3*2^{n-1}.\) in this case.
Geometric Series: Geometric series can be understood as the summation of all the terms of the geometric sequences; in other words, if the ratio between every term to its prior term is always fixed then it is stated to be a geometric series.
If \(a,\ ar,\ ar^2,\ ar^3,\dots.\ ar^{n-1}\) denotes a geometric sequence. Then the geometric series is given by the formula:
\(a+ar+ar^2+ar^3+\dots.+ar^{n-1}\).
Harmonic Sequence and Series
The third type is harmonic sequence and series. Let us understand the harmonic sequence first followed by the harmonic series.
Harmonic Sequence: A sequence of numbers is said to be in harmonic sequence if the reciprocals of all the elements/numbers/data of the sequence form an arithmetic sequence. The general equation of a harmonic sequence is as follows:
\(\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},……\frac{1}{a_n}\)
If we calculate the nth term of the Harmonic Sequences then we get \(\frac{1}{[a+(n-1)d]}\) where a represents the first term of the arithmetic sequence, d denotes the common difference of the arithmetic sequence and “n” is the number of terms/elements in the arithmetic sequence.
Example of Harmonic Sequence: \(\frac{1}{3},\frac{1}{6},\frac{1}{9},\frac{1}{12},\frac{1}{15}.\)
Here reciprocal of all the terms are in the arithmetic sequence: 3, 6, 9, 12, 15.
Also if the sequence a, b, c, d, …is assumed to be an arithmetic sequence; then the harmonic sequence can be written as:
\(\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d},\dots.\)
Harmonic Series: A series developed by applying a harmonic sequence is recognized as a harmonic series. If
\(\frac{1}{a},\ \frac{1}{a+d},\ \frac{1}{a+2d},\ \frac{1}{a+3d},\dots\) denotes a harmonic sequence then the harmonic series is given by the expression as shown below:
\(\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+\frac{1}{a+3d}+\dots\)
Example for the harmonic sequence \(\frac{1}{3},\frac{1}{6},\frac{1}{9},\frac{1}{12},\frac{1}{15}\) the corresponding harmonic series is: \(\frac{1}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\frac{1}{15}\dots\).
Series with both positive and negative elements/terms, but in a regular pattern, they alternate, as in the alternating harmonic series.
\(\sum_{n=1}^∞\frac{(−1)^{n−1}}{n}=\frac{1}{1}−\frac{1}{2}+\frac{1}{3}\ −\frac{1}{4}+⋯\)
Fibonacci Numbers
Fibonacci numbers or arithmetic are a form of a sequence of numbers in which each component is obtained by adding two preceding elements and the sequence begins with 0 and 1. The sequence is represented as,
\(F_0=0\ and\ F_1=1\ and\ F_n=F_{n-1}+F_{n-2}\)
Example of Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 11, 19, …
Difference Between Sequence and Series
So far we study the definition, various sequence, and series formulas as well as examples. Now let us check out some of the differences between sequence and series.
The following table shows the difference between sequence and series:
| Sequences | Series |
| The set of components follows a pattern in a sequence | The Sum of elements of the sequence is known as series |
| Order of components is essential | The order of components is not so significant |
| Finite sequence example: 4,5,6,7,8 | Finite series example: 4+5+6+7+8 |
| Infinite sequence example: 4,5,6,7,8,…… | Infinite Series example: 4+5+6+7+8+…… |
Sequence and Series Formulas
There are several formulas associated with sequences and series using which we can determine a set of unknown values like the first term, nth term, common difference, the sum of n terms, and other parameters. These formulas are different for specific sorts of sequences and series. Let us have a look at these sequence formulas.
Arithmetic Sequence and Series Formulas
The following table shows the difference between arithmetic sequence and series formulas:
| Term | Arithmetic Sequence and Series Formula |
| Arithmetic sequence | \(\left\{a,\ a+d,\ a+2d,\ a+3d,\dots\right\}\) |
| Arithmetic series | \(\left\{a+\left(a+d\right)+\left(a+2d\right)+\left(a+3d\right)\dots\right\}\) |
| First term | a |
| Common difference(denoted by d) | Successive element – Preceding element \(a_n−a_{n−1}\) |
| nth term/General Term(\(a_n\)) | \(a_n=a+(n-1)\ d\) |
| Sum of first n terms | \(S_n=\frac{n}{2}\times\left[2a+\left(n-1\right)d\right]\ or\ S_n=\frac{n}{2}\times\left[a+l\right]\) Where l denotes the last element of the series. |
Geometric Sequence and Series Formulas
The following table shows the difference between geometric sequence and series formulas:
| Term | Geometric Sequence and Series Formula |
| Geometric sequence | \(a,\ ar,\ ar^2,\ ar^3,\dots.\ ar^{n-1}\) |
| Geometric series | \(a+ar+ar^2+ar^3+\dots.+ar^{n-1}\) |
| First term | a |
| Common difference(denoted by r) | Successive element /Preceding element \(r=\frac{ar^{\left(n-1\right)}}{ar^{\left(n-2\right)}}\) |
| nth term/General Term (\(a_n\)) | \(a_n=ar^{n-1}\) |
| Sum of first n terms | \(\begin{array}{l}S_n=a\left(\frac{1-r^n}{1-r}\right),\ for\ \left|r\right|<1\\\ S_n=a\left(\frac{r^n-1}{r-1}\right),\ for\ \left|r\right|>1\end{array}\) If the number of terms is infinite in a GP, then the sum of terms is given by: \(S_n=\frac{a}{1-r},\ \left|r\right|<1\) |
Harmonic Sequence and Series Formulas
The following table shows the difference between harmonic sequence and series formulas:
| Term | Harmonic Sequence and Series Formula |
| Harmonic sequence | \(\frac{1}{a},\ \frac{1}{a+d},\ \frac{1}{a+2d},\ \frac{1}{a+3d},\dots\) |
| Harmonic series | \(\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+\frac{1}{a+3d}+\dots\) |
| First term | a |
| Common difference | d |
| nth term/General Term(\(a_n\)) | \(\frac{1}{[a+(n-1)d]}\) |
| Harmonic Mean ‘H’ between a and b | \(H=\frac{2ab}{a+b}\) |
| Sum of first n terms | \(\begin{array}{l}For\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\dots.,\frac{1}{a+(n-1)d}\\ S_n=\frac{1}{d}\ln\left(\frac{2a+\left(2n−1\right)d}{2a−d}\right)\end{array}\) |
Sum of n terms of Some Special Series
Here are some additional types of mathematical series formulas.
- \(\begin{array}{l}1+2+3+……..+n=\sum_{ }^{ }n=\frac{n\left(n+1\right)}{2}\\
- 1^2+2^2+3^2+……..+n^2=\sum_{ }^{ }n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\\
- 1^3+2^3+3^3+……..+n^3=\sum_{ }^{ }n^3=\left[\frac{n\left(n+1\right)}{2}\right]^2\end{array}\)
Important Points of Sequence and Series
- If a constant is added or subtracted from every term of an AP then the resulting sequence is also an AP with the same common difference.
- If the individual elements of an AP are multiplied/divided with a non-zero constant, say k, then the resulting sequence is also an AP where the common difference is given by k × d or k / d, where d indicates the common difference of the provided AP.
- In a finite AP, the sum of the element equidistant from the beginning and the ending is always the same.
- If all the terms of a GP are multiplied or divided by the same non-zero constant, then the result is also a GP with the same common ratio.
- If reciprocal of all the terms of a GP is taken, then the results also form a GP.
- If every component of a GP is raised to the identical power, then the resulting sequence also forms a GP.
- In a finite GP, the product of the elements equidistant from the origin and the ending is regularly the same and is equivalent to the product of the first term and the last term.
Sequence and Series Examples
Well acquainted with the series and sequence formulas, along with the difference between sequence and series it’s time to solve a few sequence and series examples.
Example 1. If the sequence 2, 5, 8…… is in AP and if each term of the sequence is multiplied by 3. Then the resultant sequence is in?
Solution: Given: The sequence 2, 5, 8…… is in AP, common difference d = 3 and k = 3.
Here, each term of the sequence 2, 5, 8…… is multiplied by 3.
Hence, the resultant sequence is also in AP with a common difference, K × d = 3 × 3 =9.
Example 2. What is the value of? \(7^{\frac{1}{7}}\times7^{\frac{1}{7^2}}\times7^{\frac{1}{7^3}}\times………\ upto\ \infty\)
Solution: If \(a,\ ar,\ ar^2,\ ar^3,\dots.\ ar^{n-1}\) is an infinite Geometric Progression, then the sum of infinite geometric series is given by:
\(S_n=\frac{a}{1-r},\ \left|r\right|<1\)
Calculation:
We know that the series is an infinite geometric series with the first term a = 1/7 and common ratio r = 1/7
\(\frac{\left(\frac{1}{7}\right)}{\left(1-\frac{1}{7}\right)}=\frac{1}{6} = 7^{\left(\frac{1}{6}\right)}\)
Example 3. If \( \frac{1}{4}, \frac{1}{x}, \frac{1}{10}\) are in HP, then find the value of x?
Solution : If \(a_1,\ a_2,\ a_3,\ a_4,\dots\dots\dots \) is AP then \(\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},……\frac{1}{a_n}\) is HP and vice versa.
Given sequence is \( \frac{1}{4}, \frac{1}{x}, \frac{1}{10}\) are in HP
⇒ 4, x and 10 are in AP.
⇒ x – 4 = 10 – x
⇒ x = 7
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| If you are checking Sequences and Series article, also check the related maths articles: | |
| Missing Numbers in Series and Sequence | Integral Test |
| Arithmetic Progression | Sum of n Natural Numbers |
Sequences and Series FAQs
Q.1 What is the difference between sequence and series?
Ans.1 Sequence is a particular arrangement of elements in some definite method, whereas series is the total of the elements of the sequence. In sequence order of the components are fixed, but in series, the order of components is not fixed.
Q.2 What are series and sequences used for?
Ans.2 Sequences and Series represent an important role in several aspects of our lives. Series and sequences are used in the prediction, evaluation and monitoring of the outcome of a situation or event for decision-making.
Q.3 What are the numbers in the Fibonacci sequence?
Ans.3 Fibonacci numbers are a form of a sequence of numbers in which each component is obtained by adding two preceding elements and the sequence begins with 0 and 1. The Fibonacci Sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55….
Q.4 What are finite or infinite sequences?
Ans.4 When the sequence continues with endless terms then it is named an infinite sequence, otherwise, it is a finite sequence.
Q.5 What are Some of the Common Types of Sequences?
Ans.5 There are 3 types of sequences and series:
- Arithmetic Sequence and Series
- Geometric Sequence and Series
- Harmonic Sequence and Series
