JEE Main Study Notes for Sequence and Series: A sequence is a group of numbers that obeys a certain pattern in an ordered form and by adding or subtracting the terms of sequence a series is constructed.

• In JEE Main Mathematics Question Paper candidates can expect around 1-2 questions on the topic Sequence and Series.
• The weightage of Sequence and Series in JEE Main 2022 is 10-12 marks.
• Sequence and Seies consists of topics like Sequence, Infinite Sequence, Series, General term of A.P., Arithmetic Mean, Geometric Progression or GP, General term of G.P., Geometric Mean, Harmonic Progression or H.P., General term of H.P., Harmonic Mean, etc.

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Important Topics from Sequence & Series

Below mentioned are some topics (along with questions followed in the article) based on Sequence and Series from JEE Main point of view.

What is Sequence?

A group of numbers in an ordered form that follows a certain pattern is called a Sequence. The numbers in a Sequence are called terms or elements or members and the total number of elements in a sequence is called the length of a sequence.

A sequence is similar to a set of numbers but different in the fact that in a sequence, numbers can be repeated while they cannot be repeated in the case of a set.

• {1, 3, 9, 27} is the sequence of multiples of 3.
• {m, o, n, k, e, y} is the sequence of letters in the word "monkey".
• {1, 2, 3, 4, ….. } is a very simple sequence of numbers that can be called an infinite sequence.
• {20, 25, 30, 35, ….. } is the sequence of multiples of 5 and is also called an Infinite Sequence.
• {a, b, c, d, e} is the sequence of the first 5 letters alphabetically.
• {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case).

Arithmetic Progression

Arithmetic Progression is defined as a series in which the difference between any two consecutive terms is constant throughout the series. This constant difference is called the common difference. The difference is represented by “d”.

If the first term of an arithmetic sequence is a1 and the common difference is d, then the nth term of the sequence is given by: an= a1+ (n−1) d

Sum of n terms of an arithmetic progression: Let sum be Sn

Sn= {a} + {a + d} + {a + 2d} +…+ {a + (n–1)d}

or, Sn = n/2 [2a+n-1d]

Selection of terms in AP:

1. 3 terms: a – d, a, a + d.
2. 4 terms: a – 3d, a – d, a + d, a + 3d.
3. 5 terms: a – 2d, a – d, a, a + d, a + 2d.

Example: If 1, log9 (31-x + 2), log3(4.3x – 1) are in AP, then x equals:

(1) log34

(2) 1 - log34

(3) 1 - log43

(4) log43

Solution: (2)

Must Read:

• JEE Main Study Notes for Matrices & Determinants

• JEE Main Study notes for Integral Calculus

Arithmetic Mean of Two Terms A & B

If a, A, b are in A.P. then A is called the Arithmetic Mean of numbers a and b, we get 

Let A1, A2, A3, ……, An be such that a, A1, A2, A3, ……, An,b is A.P.

Clearly

; ;

.....................,

..................... ;

We have 

That is, the Sum of n A.M. terms between a and b = n × A.M. of a and b

Geometric Progression

A sequence in which the ratio between every successive term is constant is called Geometric Progression. It could be in ascending or descending form according to the constant ratio.

Example: 1, 4, 16, 64, ….

Here, in this example,

a1= 1

a2= 4 = a1(4)

a3 = 16 = a2(4)

Here we are multiplying it by 4 every time to get the next term. Here the ratio is 4.

The ratio is denoted by “r”.

The nth term of a geometric progression is:

an = an−1⋅r

or an= a1⋅rn−1

The sum of n terms of a geometric progression is: 

Selection of terms in GP:

1. 3 terms: a/r, a, ar.
2. 4 terms: a/r3, a/r, ar, ar3.
3. 5 terms: a/r2, a/r, a, ar, ar2.

Must Read:

• JEE Main Study Notes for Mathematical Reasoning
• JEE Main Study Notes for Algebra

Example:

Geometric Mean of Two Terms A and B

If a, G, b are in G.P. (a and b are positive), then G is the Geometric Mean of numbers a and b. We get

Must Read:

• JEE Main Study Notes on Differential Calculus

• JEE Main Study Notes for Permutation and Combinations

Harmonic Progression

If a1, a2, a3, ……, an are in AP, such that none of them is zero, then (1/ a1, 1/ a2………. 1/ an) are said to be in HP.

1. If a, b, c are in HP, then (1/a), (1/b), (1/c) are in AP.
2. If a, H1, H2,………, Hn, b are in HP, then H1, H2,………,Hn are called n harmonic means between a and b.

Example:

Harmonic Mean of Two Non-Zero Terms a and b

If a, H, b are in H.P., then H is the Harmonic Mean of numbers a and b, we get

Relation between Arithmetic Mean & Geometric Mean

Here, we can see that sequences 2, 6, 18 are a geometric progression.

As we know that the formulae for the arithmetic mean and geometric mean are as follows:

Where a and b are the positive integers.

Example: Find two numbers, If the arithmetic mean and the geometric mean of two positive real numbers are 20 and 16, respectively.

Solution: 

Now we will put these values of a and b in

(a-b)2 = (a+b)2 – 4ab

(a-b)2 = (40)2 – 4(256)

= 1600 – 1024

= 576

a – b = ± 24 ( by taking the square root) …(3)

By solving (1) and (3), we get

a + b = 40

a – b = 24

a = 8, b = 32 or a = 32, b = 8.

Relations Between Arithmetic Mean, Geometric Mean, and Harmonic Mean

If a and b are positive numbers, then 

. We have

• Arithmetic Mean, Geometric Mean, and Harmonic Mean is in G.P., i.e., G2 = AH
• Arithmetic Mean, Geometric Mean, and Harmonic Mean., i.e., A≥G≥H. Equality holds if and only if a = b.
• If a1, a2, a3, ……, an be n positive numbers, then we define

A.M. 

G.M. G = (a1, a2, a3, ……, an) 1/n

H.M. 

We again have A > G > H. Equality holds if and only if a1 = a2 = …….. = an

Must Read: JEE Main Study Notes for Statistics

Special Series

Special Series is a series which is special in some way. It could be arithmetic or geometric.

Some of the special series are:

1. Sum of first n natural numbers: 1 + 2 + 3 + 4 + ……. + n
2. Sum of squares of first n natural numbers: 12+ 22+ 32+ 42+ ……. +n2
3. Sum of cubes of first n natural numbers: 13+ 23+ 33+ 43 + ……. + n3

Some standard results: 

Also Check: JEE Main Scalers and Vectors Study Notes

Practice Questions on Sequence & Series

Question: The sum of the series 1 + [1] / [4 × 2!] + [1] / [16 × 4!] + [1] / [64 × 6!] + . . . . . is ________.

Solution: [ex + e-x] / [2] = 1 + [x2/ 2!] + [x4/ 4!] + [x6 / 6!] + . . . . ∞

Putting x = 1 / 2, we get 1 + [1] / [4 × 2!] + [1] / [16 × 4!] + [1] / [64 × 6!] + . . . . . ∞

= [e1/2 + e-1/2 ] / [2]

= [e + 1] / 2√e

Question: The interior angles of a polygon are in A.P. If the smallest angle be 120∘ and the common difference be 50∘, 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 = 120∘ , d = 5, therefore

[n / 2] × [2 × 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 = 120∘ + 15.5∘ = 195∘, which is impossible as interior angle cannot be greater than 180∘.

Hence, n = 9.

Question: The sum of n terms of the following series 1 + (1 + x) + (1 + x + x2) + . . . . . will be ___________.

Solution:

1 + (1 + x) + (1 + x + x2) + . . . . .+ (1 + x + x2+ x3+ . . . + xn-1) + . . .

Required sum = [1 / (1 − x)] * {(1 − x) + (1 − x2) + (1 − x3) + (1 − x4) + ……….upto n terms}

= [1 / (1 − x)] * [n − {x + x2 + x3 + . . . . . . . . . . upto n terms } ]

= [1 / (1 − x)] * [n − {x (1 − xn) / [1 − x]}]

= [n (1 − x) − x (1 − xn)] / [(1-x)2]

Question: Let x + y + z = 15 if 9, x, y, z, a are in A.P.; while [1 / x] + [1 / y] + [1 / z] = 5 / 3 if 9, x, y, z, a are in H.P., then what will be the value of a?

Solution:

x + y + z = 15, if x = (z-3)-1 = z3 are in A.P.

Sum =9+15+a=52(9+a)

⇒ 24 + a = 5 / 2 (9 + a)

⇒ 48 + 2a = 45 + 5a

⇒ 3a = 3

⇒ a = 1 ..(i) and

[1 / x] + [1 / y] + [1 / z] = 5 / 3, if 9, x, y, z, are in H.P.

Sum = 1 / 9 + 5 / 3 + 1 / a

= 5 / 2 [1 / 9 + 1 / a]

⇒ a = 1

Tips & Tricks to Study Sequence & Series

• Go through the detailed syllabus and divide each topic according to the time left to prepare for the examination.
• Memorize some of the important formulae of the sum of some famous and important series.
• A complete numerical series is followed by an incomplete numerical series. You need to solve that incomplete numerical series in the same pattern in which the complete numerical series is given.
• Candidates can purchase the online test series from some of the best and top coaching institutes in online mode only.
• JEE Main does not require a procedure of how you arrived at the result, it just requires straight answers. Thus there is a scope for intelligent guesses. Do not rely completely on guesswork.
• Try to systematically strike down options by the process of elimination. For questions that have “All of These” as an option, this trick especially works for that.
• Candidates are advised to attempt at least 10- 15 previous year sample papers before appearing for the actual examination to understand the paper pattern, marking scheme, and types of questions asked in the examination.

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