JEE Important Chapter - Sequence and Series

Get interactive courses taught by top teachers

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.

Important Topics of Sequence and Series

• Sequences

• Arithmetic Mean

• Geometric Mean

• Arithmetic progression

• Geometric Progression

• Harmonic Progression

• Sum up to n terms

• Arithmetic-geometric series

Important Concepts of Sequence and Series

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 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.

Geometric 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.

Harmonic Sequences

A harmonic sequence is one in which each term of an arithmetic sequence is multiplied by the reciprocal of that term. A harmonic series 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.

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

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 + ...

Sequence and Series Formulas

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 termCommon 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 termCommon 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 < 1sn = a(rn -1)/(r – 1) if r > 1

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.

Solved Previous year Questions

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

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.

See More

JEE Main Important Dates

View All Dates
JEE Main 2022 June and July Session exam dates and revised schedule have been announced by the NTA. JEE Main 2022 June and July Session will now be conducted on 20-June-2022, and the exam registration closes on 5-Apr-2022. You can check the complete schedule on our site. Furthermore, you can check JEE Main 2022 dates for application, admit card, exam, answer key, result, counselling, etc along with other relevant information.
See More
View All Dates

JEE Main Information

Application Form
Eligibility Criteria
Reservation Policy
NTA has announced the JEE Main 2022 June session application form release date on the official website https://jeemain.nta.nic.in/. JEE Main 2022 June and July session Application Form is available on the official website for online registration. Besides JEE Main 2022 June and July session application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details. July Session's details will be updated soon by NTA.

JEE Main Syllabus

View JEE Main Syllabus in Detail
It is crucial for the the engineering aspirants to know and download the JEE Main 2022 syllabus PDF for Maths, Physics and Chemistry. Check JEE Main 2022 syllabus here along with the best books and strategies to prepare for the entrance exam. Download the JEE Main 2022 syllabus consolidated as per the latest NTA guidelines from Vedantu for free.
See More
View JEE Main Syllabus in Detail

JEE Main 2022 Study Material

View all study material for JEE Main
JEE Main 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Main Study Materials. It covers the entire JEE Main syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Main study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Main exam 2022 with Vedantu right on time.
See More
All
Mathematics
Physics
Chemistry
See All

JEE Main Question Papers

see all
Download JEE Main Question Papers & ​Answer Keys of 2021, 2020, 2019, 2018 and 2017 PDFs. JEE Main Question Paper are provided language-wise along with their answer keys. We also offer JEE Main Sample Question Papers with Answer Keys for Physics, Chemistry and Maths solved by our expert teachers on Vedantu. Downloading the JEE Main Sample Question Papers with solutions will help the engineering aspirants to score high marks in the JEE Main examinations.
See More

View all JEE Main Important Books
In order to prepare for JEE Main 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, RS Aggarwal Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Main 2022 exam so that they can grab the top rank in the all India entrance exam.
See More
Maths
NCERT Book for Class 12 Maths
Physics
NCERT Book for Class 12 Physics
Chemistry
NCERT Book for Class 12 Chemistry
Physics
H. C. Verma Solutions
Maths
R. D. Sharma Solutions
Maths
R.S. Aggarwal Solutions
See All

JEE Main Mock Tests

View all mock tests
JEE Main 2022 free online mock test series for exam preparation are available on the Vedantu website for free download. Practising these mock test papers of Physics, Chemistry and Maths prepared by expert teachers at Vedantu will help you to boost your confidence to face the JEE Main 2022 examination without any worries. The JEE Main test series for Physics, Chemistry and Maths that is based on the latest syllabus of JEE Main and also the Previous Year Question Papers.
See More

JEE Main 2022 Cut-Off

JEE Main Cut Off
NTA is responsible for the release of the JEE Main 2022 June and July Session cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Main 2022 June and July Session is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Main qualifying marks for 2021 ranged from 87.8992241 for general-category, while for OBC/SC/ST categories, they ranged from 68.0234447 for OBC, 46.8825338 for SC and 34.6728999 for ST category.
See More

JEE Main 2022 Results

JEE Main 2022 June and July Session Result - NTA has announced JEE Main result on their website. To download the Scorecard for JEE Main 2022 June and July Session, visit the official website of JEE Main NTA.
See More
Rank List
Counselling
Cutoff
JEE Main 2022 state rank lists will be released by the state counselling committees for admissions to the 85% state quota and to all seats in NITs and CFTIs colleges. JEE Main 2022 state rank lists are based on the marks obtained in entrance exams. Candidates can check the JEE Main 2022 state rank list on the official website or on our site.

JEE Top Colleges

View all JEE Main 2022 Top Colleges
Want to know which Engineering colleges in India accept the JEE Main 2022 scores for admission to Engineering? Find the list of Engineering colleges accepting JEE Main scores in India, compiled by Vedantu. There are 1622 Colleges that are accepting JEE Main. Also find more details on Fees, Ranking, Admission, and Placement.
See More

FAQs on JEE Important Chapter - Sequence and Series

FAQ

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 +…..