JEE Important Chapter - 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 Chapter-wise Solutions 2022-23 

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 and Series Formulas

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

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

Answer: 1-D, 2-B

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.

Important Related Links for JEE Main 2022-23