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JEE Main Important Chapter - Sequence and Series

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Last updated date: 16th May 2024
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JEE Main: What are Sequence and Series?

One of the most fundamental subjects in Arithmetic is sequence and series. A sequence is an itemized collection of elements that allows for any type of repetition, whereas a series is the sum of all elements. An arithmetic progression is one of the most common examples of sequence and series.


By answering problems based on the formulas, the principles can be better grasped. They are similar to sets, but the main difference is that individual terms in a sequence might appear multiple times in different positions. The length of a sequence is equal to the number of terms it contains, and it might be finite or infinite. The ideas of sequence and series will be discussed here with the use of definitions, formulas and examples.


JEE Main Maths Chapters 2025


Important Topics of Sequence and Series Chapter

  • Sequences

  • Arithmetic Mean

  • Geometric Mean

  • Arithmetic progression

  • Geometric Progression

  • Harmonic Progression

  • Sum up to n terms

  • Arithmetic-geometric series


Maths Sequence and Series Important Concepts For JEE Main

Sequence and series include various important topics from which questions come directly or indirectly in exams. Let’s discussed all the topics one-by-one in detail. 


What is Sequence and Series?

A sequence is an arrangement of any objects or a set of numbers in a particular order(manner) followed by some rule. If a1, a2, a3, a4,……… etc. denotes the term of a sequence, then 1, 2, 3, 4, ….. denotes the position of the term.

A sequence can be defined based on the number of terms, which is either a finite sequence or infinite sequence.

Series is a sum of elements that follow a pattern. For example, if a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by Sn = a1+a2+a3 + .. + an.

 

Types of Sequence and Series

Some of the common types of sequence examples are

  • Arithmetic Sequences

  • Geometric Sequences

  • Harmonic Sequences

  • Fibonacci Numbers


Arithmetic Progression Sequences

An arithmetic sequence is one in which each term is either the addition or subtraction of a common term known as the common difference. An arithmetic sequence is, for example, 1, 4, 7, 10,... The arithmetic series is a series constructed by applying an arithmetic sequence. Consider, for example, 1 + 4 + 7 + 10... is an arithmetic series.

 Nth term of an AP: The nth term of an AP can be found using the following formula:

$a_n = a + d(n - 1)$


Geometric Progression Sequences

A geometric sequence is a sequence where the successive terms have a common ratio, which is obtained by multiplying or dividing a definite number with the preceding number. For example, 1, 4, 16, 64, ...is a geometric sequence. A series formed by using a geometric sequence is known as the geometric series. For example 1 + 4 + 16 + 64... is a geometric series. The geometric progression can be of two types: Finite geometric progression and an infinite geometric series.

 Nth term of a GP: The nth term of a GP can be found using the following formula:

an = ar^(n - 1)


Harmonic Progression Sequences

A harmonic sequence is one in which each term of an arithmetic sequence is multiplied by the reciprocal of that term. A harmonic sequence is 1, $\dfrac{1}{4}$, $\dfrac{1}{7}$, $\dfrac{1}{10}$,..... and so on. The harmonic series is a series constructed by employing a harmonic sequence. For example, 1 + 1/4 + 1/7 + 1/10.... is a harmonic series.

Nth term of an HP: The nth term of an HP can be found using the following formula:

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


Fibonacci Numbers

Fibonacci numbers are a fascinating number series in which each element is created by adding two preceding elements, with the sequence beginning with 0 and 1. A sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn-1 + Fn-2

F1=0, 

F2 = 1, 

F3 = 1,

F4 = 2,

F5 = 3,

..

Fn = Fn-1 + Fn-2


Series

A series is the sum of the terms of a sequence. Series can be finite or infinite. A finite series is the sum of a finite number of terms, while an infinite series is the sum of an infinite number of terms.

For example, if a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by Sn = a1+a2+a3 + .. + an.


Types of Series

Arithmetic series: An arithmetic series is the sum of the terms of an arithmetic sequence. For example, the sum of the first n terms of an arithmetic sequence with first term a and common difference d is given by the formula:

$S_n = \dfrac{n}{2}(2a + (n - 1)d)$


Geometric series: A geometric series is the sum of the terms of a geometric sequence. For example, the sum of the first n terms of a geometric sequence with first term a and common ratio r is given by the formula:

$S_n = \dfrac{a(1 - r^n)}{(1 - r)}$


Harmonic series: A harmonic series is the sum of the terms of a harmonic sequence. For example, the sum of the first n terms of a harmonic series is given by the formula:

$S_n = H_n$ + γ

where $H_n$ is the nth harmonic number and γ is the Euler-Mascheroni constant.



Arithmetic Mean

The arithmetic mean is the average of two numbers. If we have two numbers n and m, then we can include a number A in between these numbers so that the three numbers form an arithmetic sequence, like n, A, m.

 

In that case, the number A is the arithmetic mean of the numbers n and m.


Arithmetic Mean can be used to calculate the central tendency or the approximate centre point of an arithmetic sequence. 

 

Arithmetic mean for an arithmetic sequence can be calculated using the below formula.

 

A = (n+m)/2

 

Geometric Mean

Geometric Mean is the average of two numbers in a geometric sequence. If p and q are the two numbers of the sequence, then the geometric mean will be 

 

\[GM = \sqrt{pq}\]

 

Geometric mean, similar to arithmetic mean, is used to calculate the central tendency or the approximate mid element of any given geometric sequence.

 

Harmonic Mean

By the harmonic mean definition, harmonic mean is the reciprocal of the arithmetic mean, the formula to define the harmonic mean “H” is given as follows: 

Harmonic Mean(H) = \[n/[(1/x_{1})+(1/x_{2})+(1/x_{3})......(1/x_{n})]\]

 Where

n is the total number of terms and

 x1, x2, x3,…, xn are the individual values up to nth terms.


Arithmetic-Geometric Series

An arithmetic-geometric series (AGS) is a sequence in which each term is the product of the corresponding terms of an arithmetic sequence and a geometric sequence.

For example, the following sequence is an AGS:

$1 + 3x + 6x^2 + 9x^3 + 12x^4 + 15x^5 + 18x^6 + …$

The arithmetic sequence is 1, 3, 6, 9, 12, 15, 18, ... and the geometric sequence is $x, x^2, x^3, x^4, x^5, x^6, ....$

The general form of an AGS is: $a(1 + r)^(n - 1)$

where a is the first term, r is the common ratio, and n is the term number.

Sum of an AGS: $S_n = \dfrac{a(1 - r^n)}{(1 - r)}$

where a is the first term, r is the common ratio, and n is the number of terms.


Arithmetico-Geometrical Progression 

An arithmetic-geometrical progression (AGP) is a sequence of numbers in which each term is the product of a term of an arithmetic progression and a term of a geometric progression.

In other words, an AGP is a sequence of the form:

$a_1, a_1r + d, a_1r^2 + 2d, a_1r^3 + 3d, …$

The sum of the first n terms of an AGP is given by the formula:

$S_n = \dfrac{a(1 - r^n)}{(1 - r) + nd}$

The nth term of an AGP is given by the formula:

$a_n = a_1r^(n - 1) + (n - 1)d$


Difference Between Sequence and Series

Some of the common differences between sequence and series are:

Sequence

Series

In sequence, elements are placed in a particular order following some of the 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 in sequence.

The order of appearance is not important in the series. 

Example: Harmonic sequence is 1, 1/2, 1/3, 1/4, ...

Example: Harmonic series is 1 + 1/2 + 1/3 + 1/4 + ...

 

List of Important Formulas For Maths Sequence and Series Chapter

Some of the arithmetic progression and geometric progression formulas are given below:

Sl. No.

Name of the Concept 

List of Formulas

1

Arithmetic progression Sequence

a, a+d, a+2d,……,a+(n-1)d,….

2

AP Common Difference or Ratio

Successive term – Preceding term

Common difference(d) = a2 – a1

3

AP General Term (nth Term)

an = a + (n-1)d

4

AP nth term from the last term

an = l – (n-1)d

5

AP Sum of first n terms

sn = n/2(2a + (n-1)d)

6

Geometric progression Sequence

a, ar, ar2,….,ar(n-1),…

7

GP Common Difference or Ratio

Successive term/Preceding term

Common ratio = r = ar(n-1)/ar(n-2)

8

GP General Term (nth Term)

an = ar(n-1)

9

GP nth term from the last term

an = 1/r(n-1)

10

GP Sum of first n terms

sn = a(1 – rn)/(1 – r) if r < 1

sn = a(rn -1)/(r – 1) if r > 1

 

Number Series Formula For JEE Main

Number series is a crucial topic in the quantitative aptitude section of JEE Main. A solid grasp of the underlying formulas is essential for solving problems efficiently. Here's a breakdown of the key formulas:


Arithmetic Series:

$\text{Sum of an arithmetic series:} \quad S_n = \frac{n}{2}[2a + (n-1)d] \\

\text{\(n^{th}\) term of an arithmetic series:} \quad a_n = a + (n-1)d$


Geometric Series:

$\text{Sum of a geometric series:} \quad S_n = \frac{a(r^n - 1)}{r-1} \\

\text{\(n^{th}\) term of a geometric series:} \quad a_n = ar^{(n-1)}$


Special Series:

$\text{Sum of the squares of the first ‘n’ natural numbers:} \quad S_n = \frac{n(n+1)(2n+1)}{6} \\

\text{Sum of the cubes of the first ‘n’ natural numbers:} \quad S_n = \left[\frac{n(n+1)}{2}\right]^2$


Fibonacci Series:

$F_n = F_{n-1} + F_{n-2} \quad \text{with initial values } F_0 = 0 \text{ and } F_1 = 1$


Sum of Series with Common Differences:

$S_n = n^2 \quad \text{(Sum of the series \(1 + 3 + 5 + \ldots + (2n-1)\))}$


Sum of Squares of Consecutive Numbers:

$S_n = \frac{n(n+1)(2n+1)}{6} \quad \text{(Sum of the series \(1^2 + 2^2 + 3^2 + \ldots + n^2\))}$


Miscellaneous Series:

Various series involving alternate signs, squares, cubes, etc., should be approached with specific formulas depending on the pattern.


Understanding these formulas equips aspirants with the tools needed to tackle diverse number series problems in JEE Main. Regular practice and application of these formulas in problem-solving scenarios will enhance both speed and accuracy, contributing to a strong performance in the examination. Remember, mastery over these formulas is not just about memorization but about applying them intelligently to solve a wide range of problems.


Summary Points

  • In a math series and sequence, a is considered to be the first term, d is the common difference, and an is known to be the nth term. 

  • The arithmetic sequence can be explained as a, a + d, a + 2d, a + 3d, …

  • Each term in a geometric progression is obtained by multiplying the common ratio of the successive term to the preceding term.

  • The geometric progression formula is known as :

an = arn-1 

  • The sum of infinite geometric progression formula is known as :

Sn = a/(1-r) Where |r| < 1.


JEE Main Maths Sequence and Series Solved Examples

Question 1: If 4,7,10,13,16,19,22……is a sequence, Find:

a. Common difference

b. nth term

c. 21st term

Solution: Given sequence is, 4,7,10,13,16,19,22……

a) The common difference = 7 – 4 = 3

b) The nth term of the arithmetic sequence is denoted by the term Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d, is the

common difference.

Tn = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1

c) 21st term as:  T21 = 4 + (21-1)3 = 4+60 = 64.

 

Question 2: Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term.

Solution: The common ratio (r)  = 4/1 = 4

The preceding term is multiplied by 4 to obtain the next term.

The nth term of the geometric sequence is denoted by the term Tn and is given by Tn = ar(n-1)

where a is the first term and r is the common ratio.

Here a = 1, r = 4 and n = 9

So, 9th term is can be calculated as T9 = 1* (4)(9-1)= 48 = 65536.

 

Previous year Questions From JEE Main 

Question 1: If the pth term of an A.P. be q and qth term is p, then its rth term will be __________.

Solution:

Given that, Tp = a + (p − 1)d = q …….(i) and

Tq = a + (q − 1)d = p ……. (ii)

From (i) and (ii), we get d = [−(p − q)] / [(p − q)] = −1

Putting value of d in equation (i), then a = p + q − 1

Now, rth term is given by A.P. Tr = a + (r − 1)d

= (p + q − 1) + (r − 1) (−1)

= p + q − r

 

Question 2: The interior angles of a polygon are in A.P. If the smallest angle be 120o and the common difference be 5o, then the number of sides is __________.

Solution:

Let the number of sides of the polygon be n.

Then the sum of interior angles of the polygon = (2n − 4) [π / 2] = (n − 2)π

Since the angles are in A.P. and a = 120o, d = 5, therefore

(n / 2)  (2 $\times$ 120 + (n − 1)5) = (n − 2)  180

n2 − 25n + 144 = 0

(n − 9) (n − 16) = 0

n = 9, 16

But n = 16 gives

T16 = a + 15d = 120o + 15.5o = 195o, which is impossible as interior angle cannot be greater than 180o.

Hence, n = 9.

 

Question 3: If x, 1, z are in A.P. and x, 2, z are in G.P., then x, 4, z will be in __________.

Solution:

x, 1, z are in A.P., then

2 = x + z ……(i) and

4 = xz ……(ii)

Divide (ii) by (i), we get

$[$xz$]$ / $[$x + z$]$ = 4 / 2 or

2xz / $[$x + z$]$ = 4

Hence, x, 4, z will be in H.P.

 

Practise Questions 

1. The sum to infinity of the series $1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+\ldots .$ is :-

(A) 4

(B) 6

(C) 2

(D) 3

 

2. A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months, his savings increased by Rs. 40 more than the saving of immediate previous month. His total saving from the start of service will be Rs. 11040 after 

(A) 20 months

(B) 21 months

(C) 18 months

(D) 19 months

 

Answer: 1-D, 2-B


JEE Main Maths - Sequences and Series Study Materials

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JEE Main Maths Study and Practice Materials

Explore an array of resources in the JEE Main Maths Study and Practice Materials section. Our practice materials offer a wide variety of questions, comprehensive solutions, and a realistic test experience to elevate your preparation for the JEE Main exam. These tools are indispensable for self-assessment, boosting confidence, and refining problem-solving abilities, guaranteeing your readiness for the test. Explore the links below to enrich your Maths preparation.



Conclusion

As per the article, there are a variety of formulas connected to various sequences and series that can be used to determine a set of unknown values such as the first term, nth term, common parameters, and so on. Each type of sequence and series has its own set of formulas. We also went through some of the problems to help us understand them better. Practising more and more problems will help you to solve the question in exams with accuracy and speedily.

FAQs on JEE Main Important Chapter - Sequence and Series

1. What is the weightage of sequence and series in JEE main?

Algebra is a fascinating subject at the JEE level. All of the topics are more or less independent of themselves. Sequences and Series is an interesting and significant topic, and every year you will get 1 - 2 questions on it in the JEE Main exam as well as other engineering entrance examinations, as the chapter's weightage of 6.6 per cent. 

2. How to prepare for sequence and series?

Begin with the fundamentals: learn all of the definitions for sequences, series, and arithmetic and geometric progression. Remember typical results and derive and understand the equations for General Term, Sum of the Series of n terms. Learn about the notion of Harmonic Sequences and the word Harmonic Sequences in general.


Calculate all the summation equations for some special series, such as the sum of the first n natural numbers, the sum of odd numbers, the sum of the cube of the first n natural numbers, and so on. After studying key sections/topics, make sure to solve questions relating to those concepts without consulting the solutions, practise MCQ from your textbook, as well as solve all of the previous year's problems asked in JEE.

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 the case of an infinite series, the number of elements is not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +…..