Mathematics Sequences And Series Pdf

In simple words, a sequence is defined as the arrangement of objects successively and the sum of all the terms in the sequence is termed as a series. With this article on sequence and series learn about sequence definition, sequence and series formulas, along with types of series and sequence.

Sequence and series contribute a major part of mathematics, where the arrangement of objects or items in a progressive manner is termed as a sequence and the sum of all the terms in the particular sequence is termed as a series(possessing a definite relationship among all the terms/objects of the sequence).

If you are reading Sequence and Series then also go through the Number System.

What is a Sequence?

A sequence is an organisation of any objects/elements/set of digits in a particular order accompanied by some rule.

For example, 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.

Infinite or Finite

When the sequence continues with endless terms then it is named an infinite sequence, otherwise, it is a finite sequence.

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.

Similarly 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 if the sequence is finite or infinite.

Check out this article on Probability.

Types of Sequences 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 are separated by a comma and placed around curly brackets.

For example {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. Some of the popular types of sequences are as follows:

Arithmetic Sequences
Geometric Sequences
Harmonic Sequences
Fibonacci Numbers

Let's look at each of them one by one to have a detailed analysis, Starting with arithmetic sequences.

Learn the concepts of Three Dimensional Geometry here.

Arithmetic Sequences

Any sequence is termed an arithmetic sequence if the difference between the term and its previous term is always the same.

4, 8, 12, 16, 20, 24, 28……..

In the above example, the difference between the successive terms is 4.

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 are:
\(x_n=a+(n-1)\ d\)

Types of Arithmetic Sequences

There are two types of arithmetic sequence: finite and infinite sequence.

Finite sequences hold countable terms and do not progress up to infinity. An example of a finite arithmetic sequence is 4, 8, 12, 16.

An infinite arithmetic sequence is a sequence in which the elements move up to infinity.

An example of an infinite arithmetic sequence is 4, 8, 12, 16……

A series created by applying an arithmetic sequence is recognised as the arithmetic series.

For example, if 1, 3, 5, 17, …is an example of an arithmetic sequence then 1+ 3+ 5+ 17+… is the corresponding arithmetic series.

Geometric Sequences

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.

An example of geometric sequences are:

3, 6, 12, 24, 48.

Learn the various types of Relations and Functions.

Types of Geometric Sequences

Finite geometric sequence: In a finite geometric sequence, the terms are finite, for example, 32, 16, 8, 4, 2.

Infinite geometric sequence: As the name suggests, in infinite geometric sequences there are infinite terms for example 3, 6, 12, 24, 48…..

If 1, 4, 16, 64, 256…is a geometric sequence then the series formed by applying a geometric sequence is recognised as the geometric series for example 1 + 4 + 16 + 64+256… is a geometric series.

Harmonic Sequences

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.

Harmonic sequence

\(\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},……\frac{1}{a_n}\)

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:

\(\frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d},\dots.\)

The nth term of the Harmonic Sequences (H.S) =

\(\frac{1}{[a+(n-1)d]}\)

Where:

a represents the first term of the arithmetic sequence.

d denotes the common difference of the arithmetic sequence.

"n" is the number of terms/elements in the arithmetic sequence.

A series developed by applying a harmonic sequence is recognised as the harmonic series for example
\(\frac{1}{3}+\frac{1}{6}+\frac{1}{9}+\frac{1}{12}+\frac{1}{15}\dots\) is a harmonic series.

Also, read about Statistics here.

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 Fibonacci Sequence 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. Sequence is represented as, \(F_0=0\ and\ F_1=1\ and\ F_n=F_{n-1}+F_{n-2}\)

An example of Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 11, 19, …

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	\(\left\{a,\ a+d,\ a+2d,\ a+3d,\dots\right\}\)
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.

Learn more about Logarithmic functions here.

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	\(\frac{1}{a},\ \frac{1}{a+d},\ \frac{1}{a+2d},\ \frac{1}{a+3d},\dots\)
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	\(\begin{array}{l}For\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\dots.,\frac{1}{a+(n-1)d}\\ S_n=\frac{1}{d}\ln\left(\frac{2a+\left(2n−1\right)d}{2a−d}\right)\end{array}\)

Check out this article on Limit and Continuity.

Sum of n Terms of Some Special Series

Here are some additional types of mathematical series formulas.

\(\begin{array}{l}1+2+3+……..+n=\sum_{ }^{ }n=\frac{n\left(n+1\right)}{2}\\
1^2+2^2+3^2+……..+n^2=\sum_{ }^{ }n^2=\frac{n\left(n+1\right)\left(2n+1\right)}{6}\\
1^3+2^3+3^3+……..+n^3=\sum_{ }^{ }n^3=\left[\frac{n\left(n+1\right)}{2}\right]^2\end{array}\)

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.

Sequences	Series
The set of components follow 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+……

Also, read about Permutations and Combinations here.

Sequence and Series Solved Examples

Well acquainted with the series and sequence formulas, along with the difference between sequence and series it's time to solve a few 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.

Calculation:

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

Also, read about Linear Inequations here.

Sequence and Series: Key Takeaways

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.

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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 As we discussed throughout the article, Sequences and Series represents an important role in several aspects of our lives. For example in the prediction, evaluation and monitoring 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 does a Sequence and a Series Mean?

Ans.4 The set of components following a pattern is said to be a sequence whereas the sum of elements of the sequence is known as series.

Q.5 What are Some of the Common Types of Sequences?

Ans.5 Arithmetic Sequences
Geometric Sequences
Harmonic Sequences
Fibonacci Numbers

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