Sequences and Series - Topics, Formula, Books, FAQs

Consider listing the number of pages read from a book, then the list of number of pages after you read each day will result in a sequence 5,10,15,20,25,30... In Mathematics, a sequence is a list of items with some common pattern while a series is the sum of the sequence of numbers.

This article is about the concept of Sequence and Series Class 11. Sequence and Series chapter is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, VITEEE, BCECE, and more.

Sequence and Series

Almost everything in nature, from the arrangement of flower petals, patterns in animal skins, branching in plants and trees, snowflakes, honeycombs to even human DNA fragments occurs in patterns. Apart from nature, sequence and series have many real-life applications such as computer algorithms, construction, architecture, finance, etc. Sequence and series is one of the fundamental concepts for various domains like calculus, finance, physics, etc. Now, let us look into the basic concepts of sequence and series.

Sequence

A sequence is formed when terms are written in order such that they follow a particular pattern. A sequence can have any number of terms which can be finite or infinite.

The total number of terms is called the length of the sequence.

E.g., $1, 2, 3, 4, 5,..... $

$1, 4, 9, 16, ......$

$\frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6},.......$

Series

If we add or subtract all the terms of a sequence we will get an expression, which is called a series. A series is the sum of the terms of a sequence and It is denoted by $S_{n}$

If the sequence is $a_1, a_2, a_3, \ldots \ldots, a_n$, then it's sum i.e. $a_1+a_2+a_3+\ldots \ldots \ldots . .+a_n$ is a series.
$
\mathrm{S}_{\mathrm{n}}=a_1+a_2+a_3+\ldots \ldots \ldots . .+a_n=\displaystyle \sum_{\mathrm{r}=1}^{\mathrm{n}} a_r=\sum a_r
$

Sequence and Series Notes

Sequence and Series notes include the types of sequence and series.

Types of Sequence

Based on the number of terms, there are two types of sequences:

1. Finite sequence
2. Infinite Sequence

Finite Sequence

If the sequence has only a finite number of terms, then the sequence is called a finite sequence. Eg, $2, 4, 6, 8$

Infinite Sequences

If a sequence has three dots at the end, then it indicates the list never ends. It has an infinite number of terms. Then, the sequence is said to be an infinite sequence.

$
\text { E.g., } 2,4,6,8,10, \ldots
$

Based on the characteristics of the terms, the types of sequences are,

1. Arithmetic Progression
2. Geometric Progression
3. Harmonic Progression

Arithmetic Progression

An arithmetic progression is defined as the sequence in which there is same difference between every two consecutive terms. Here this difference is known as the common difference, usually denoted by ‘d’.

The general sequence of arithmetic progression is given by:

$a,a+d,a+2d,.....$

Terms in Arithmetic Progression

First term ($a_1$ (or) $a$):

As the name suggests, the first term of an AP is the first number of the progression. It is usually represented by $a_1$ (or) $a$. For example, in the AP 5, 10, 15, 20,….. the first term $a$ is 5.

Common difference (d):

In Arithmetic Progression, each term, except the first term, is obtained by adding a fixed number to its previous term. This fixed term is called the common difference. It is usually denoted by $d$. If $a$ is the first term of an AP and $d$ is the common difference, then the AP will be:

$a, (a+d), (a+2d), (a+3d), (a+4d),....$

For example, $4, 10, 16, 22, …..$ is an AP where the first term is 4 and the common difference is

$(10 - 6) = (16 - 10) = (22 - 16) = 6$.

Arithmetic progression Examples

1. For the sequence $2,5,8,11,14,17,....$ Calculate the first term as well the common difference.

The first term for the above sequence is $a=2$ and the common difference is $d=5-2=3$.

2. Examine if the sequence $4,10,16,22,28,....$ is in arithmetic progression or not?

The common difference is $d=10-4=16-10=22-16=6$. Since the difference is same, for every two consecutive terms, so the sequence is in an arithmetic progression.

Geometric Progression

A geometric progression or geometric sequence is a sequence where the first term is non-zero and the ratio between consecutive terms is always constant. The ‘constant factor’ is called the common ratio and is denoted by ‘$r$’. $r$ is also a non-zero number.

The first term of a G.P. is usually denoted by '$a$'.

If $a_1, a_2, a_3 \ldots . a_{n-1}, a_n$ is in geometric progression then, $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}=\ldots . .=\frac{a_n}{a_{n-1}}$

Eg,
$2,6,18,54, \ldots .(a=2, r=3)$
$4,2,1,1 / 2,1 / 4, \ldots .(a=4, r=1 / 2)$
$-5,5,-5,5, \ldots \ldots .(a=-5, r=-1)$

Harmonic Progression

A Harmonic Progression (HP) is defined as a sequence of real numbers obtained by taking the reciprocals of an Arithmetic Progression that excludes 0.

A sequence $a_1, a_2, a_3, \ldots ., a_n, \ldots$ of non-zero numbers is called a harmonic progression if the sequence $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots, \frac{1}{a_n}, \ldots$.

OR
Reciprocals of arithmetic progression is a Harmonic progression.
E.g. $\frac{1}{2}, \frac{1}{5}, \frac{1}{8}, \frac{1}{11}, \ldots .$. is an HP because their reciprocals $2,5,8,11, \ldots$ form an A.P.

• No term of the H.P. can be zero.

• The general form of HP is

$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2 d}, \frac{1}{a+3 d} \ldots$

Here $a$ is the first term and d is the common difference of corresponding A.P.

Types of Series

Based on the number of terms, there are two types of series:

1. Finite series
2. Infinite Series

Finite Series

If the sequence has only a finite number of terms, then the series is called a finite sequence. Eg, $2+4+6+8$

Infinite Series

If a sequence has three dots at the end, then it indicates the list never ends. It has an infinite number of terms. Then, the series is said to be an infinite series.

$
\text { E.g., } 2+4+6+8+10+\ldots
$

Based on the characteristics of the terms, the types of series are,

1. Arithmetic Series
2. Geometric Series
3. Harmonic Series

Arithmetic Series

The sum of terms of an arithmetic sequence is called the arithmetic series. The sum of the first $n$ terms of an arithmetic series is $S_n=\frac{n}{2}(2 a+(n-1) \cdot d)$ where $a$ is the first term and $d$ is the common difference.

Geometric Series

The sum of terms of a geometric sequence is called the geometric series. The sum of the first $n$ terms of a geometric series is $S_n=a \cdot \frac{1-r^n}{1-r} \quad($ for $r \neq 1)$ where $a$ is the first term and $d$ is the common difference.

Harmonic Series

The sum of the terms in a harmonic sequence is called the harmonic series. The sum of the first $n$ terms of a harmonic series is $S_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$.

Sequence and Series Formulas

Sequence and series class 11 formulas includes the terms of the sequences and the sum of first n terms of the series.

Sequence Formulas

Sequence formulas include formulas of arithmetic progression, geometric progression and harmonic progression.

Arithmetic Progression

In Arithmetic Progression (AP), several terms and notations are commonly used to describe and calculate the sequence. These terms and notations are as follows:

The number of terms (n): As the name suggests, the number of terms of an AP is the total number of terms present in the progression or in the sequence. It is usually denoted as $n$.

For example, $1, 2, 3, 4, 5$ is in AP where the number of terms ($n$) is $5$.

The general term of an AP ($t_n$): The general term or the nth term of an arithmetic sequence can be expressed in two ways i.e. the $n$th term of an AP from the beginning and the nth term of an AP from the last. We will discuss these two ways below.

The nth term of an AP from the beginning: The general term of the nth term from the beginning of an AP where the first term is $a$, the common difference is $d$, and the number of terms is $n$ is given by the formula:

$t_n = a + (n - 1)d$.

The nth term of an AP from the last: The nth term of an AP from the last, where the last term is $l$, the common difference is $d$, and the number of terms is $n$, is given by the formula:

$t_n = l - (n - 1)d$.

The nth term of an AP if the mth term is given but the first term is not given: The mth term of an AP is given.

So, $t_m=a+(m-1)d$, where $a$ is the first term and $d$ is the common difference.

Since the first term is not given, we need to find the first term in terms of the given mth term.

So, $a = t_m - (m-1)d$

Now, for the nth term,

$t_n = a+(n-1)d = t_m - (m-1)d + (n-1)d = t_m - md + d + nd - d = t_m + (n-m)d$

Therefore, $t_n = t_m + (n - m)d$ is the required nth term of an AP when the mth term is given but the first term is not given.

Arithmetic Mean

The arithmetic mean of a set of $n$ numbers is calculated by summing all the numbers and then dividing by $n$. For a set of $n$ positive integers $a_1, a_2, a_3, a_4$, $\dots$ $a_n$.

Arithmetic mean $=\frac{a_1+a_2+a_3+a_4+\ldots \ldots .+a_n}{n}$

For example, AM of 2, 4, 6, 8, 10 is $\frac{2 + 4 + 6 + 8 + 10}{5} = 6$.

Geometric Progression

General Term of a GP: If ' $a$ ' is the first term and ' $r$ ' is the common ratio, then
$
\begin{aligned}
& a_1=a=a r^{1-1}\left(1^{\text {st }} \text { term }\right) \\
& a_2=a r=a r^{2-1}\left(2^{\text {nd }} \text { term }\right) \\
& a_3=a r^2=a r^{3-1}\left(3^{\text {rd }} \text { term }\right) \\
& \cdots \\
& \cdots \\
& a_n=a r^{n-1}\left(\mathrm{n}^{\text {th }} \text { term }\right)
\end{aligned}
$

So, the general term or $\mathrm{n}^{\text {th }}$ term of a geometric progression is $a_n=a r^{n-1}$

Geometric Mean

The geometric mean of $n$ numbers is the $n$th root of the product of the numbers.

For a set of $n$ positive integers $a_1, a_2, a_3, a_4$,$ $\dots\mathrm{a}_{\mathrm{n}}$

Geometric mean $=\sqrt[n]{a_1 \times a_2 \times a_3 \times a_4 \times \ldots \ldots . \times a_n}$

Example: Geometric mean of $2, 4, 8,$ and $16=$ $\sqrt[4]{2 \times 4 \times 8 \times 16}=\sqrt[4]{1024}=5.66$

Harmonic Progression

The general term of a Harmonic Progression: The nth term or general term of an H.P. is the reciprocal of the nth term of the corresponding A.P. Thus, if $a_1, a_2, a_3, \ldots \ldots, a_n$ is an H.P. and the common difference of corresponding A.P. is d, i.e. $d=\frac{1}{a_n}-\frac{1}{a_{n-1}}$, then the nth term of corresponding AP is $\frac{1}{a_1}+(n-1) d$, and hence, the
general term or nth term of an H.P. is given by

$
a_n=\frac{1}{\frac{1}{a_1}+(n-1) d}
$

Harmonic Mean

The Harmonic Mean is the reciprocal of the average of the reciprocals of a given set of numbers. For a set of $n$ positive integers $a_1, a_2, a_3, a_4$, $\dots$ .$a_n$

$
\text { Harmonic mean }=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\frac{1}{a_4}+\ldots \ldots . .+\frac{1}{a_n}}
$

Series formulas

Series formulas include the sum of first n terms in the arithmetic series, geometric series and the harmonic series.

Sum of n terms in arithmetic series: The sum of the first $n$ terms of an arithmetic series is $S_n=\frac{n}{2}(2 a+(n-1) \cdot d)$ where $a$ is the first term and $d$ is the common difference.

Sum of n terms in geoemtric series: The sum of the first $n$ terms of a geometric series is $S_n=a \cdot \frac{1-r^n}{1-r} \quad($ for $r \neq 1)$ where $a$ is the first term and $d$ is the common difference.

Sum of n terms in harmonic series: There is no general formula for the sum of first $n$ terms that are in H.P. The sum of the first $n$ terms of a harmonic series is $S_n=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$.

Difference Between Sequence and Series

The difference between sequence and series are,

List of Topics According to NCERT/JEE MAIN

Importance of Class 11 Sequence and Series

Algebra at the JEE level is very interesting. All topics are more or less independent of each other. And one of the interesting and important topics is class 11 maths Sequences and Series and every year you will get 1-2 question in JEE Main exam as well as in other engineering entrance exams. JEE question paper is highly unpredictable, you never know questions from which topic will be asked. A general trend noticed in Mathematics paper is that a question involving multiple concepts are asked. As compared to other chapters in maths, Sequence and Series JEE MAINS questions requires less effort to prepare for the examination.

How to Study Class 11 Sequence and Series?

Sequences and series is one of the easiest topics, you can prepare this topic without applying many efforts. Start with basic theory, understand all the definition of the Sequences, series, and Arithmetic and geometric progression. Learn all the sequence and series formulas class 11 and remember standard results. Learn the concept behind Harmonic sequences and general term of Harmonic sequences. Practice many problems from each topic for better understanding. Practice from the previous year sequence and series questions.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

Important Books for Sequence and Series

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Arihant Algebra Textbook by SK Goyal or Cengage Algebra Textbook by G. Tewani but make sure you follow any one of these not all. Sequence and Series are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.