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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.

Sequences

Arithmetic Mean

Geometric Mean

Arithmetic progression

Geometric Progression

Harmonic Progression

Sum up to n terms

Arithmetic-geometric 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.

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.

Some of the common types of sequence examples are

Arithmetic Sequences

Geometric Sequences

Harmonic Sequences

Fibonacci Numbers

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.

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.

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 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

Some of the common differences between sequence and series are:

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

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.

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.

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

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.

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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 +…..

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