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    JEE Main Study Notes for Sequence and Series: Basic Concepts, Tricks & Practice Questions

    Urvashi Singhal Urvashi Singhal
    Exams Prep Master

    Sequence and Series is the easiest part of calculus in JEE Main Paper and there is no doubt that the topic is scoring too. The topic consists of 1-2 questions that totals to 10 marks in the overall marks distribution of the test. Some of the topics are Sequence, Infinite Sequence, Series, Arithmetic Progression or AP, General term of A.P., Arithmetic Mean, Geometric Progression or GP, General term of G.P., Geometric Mean, Harmonic Progression or HP, General term of H.P., Harmonic Mean, etc. These topics are quite fascinating but involve concepts which must be understood properly. It is the most scoring part of the JEE Main Paper.

    • Minimum of 1-2 questions will be asked in the JEE Main Paper from Sequence and Series.
    • The weightage of the marks of Sequence and Series in JEE Main is 10- 12 marks.

    Read the article to refer to study notes for Sequence & Series that will help you in JEE Main 2020 Preparation.

    Must Read:

    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.

    SequenceInfinite Sequence
    SeriesArithmetic Progression or AP
    General term of A.P.Arithmetic Mean
    Geometric Progression or GPGeneral term of G.P.
    Geometric MeanHarmonic Progression or HP
    General term of H.P.Harmonic Mean

    What is Sequence?

    What is Sequence?

    A group of numbers in an ordered form which follow a certain pattern is called Sequence. It is simply an ordered list of elements and can be considered as an enumerated collection of objects in which repetitions are allowed and order does not matter at all. The numbers in 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 which can be called as an infinite sequence.
    • {20, 25, 30, 35, ….. } is the sequence of multiples of 5 and are also called as 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

    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)

    Geometric Progression

    Geometric Progression

    A sequence in which the ration between every successive term is constant, it 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 with 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−1r

    or an = a1rn−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.

    Example:

    http://www.jagranjosh.com/imported/images/E/Articles/sequencesandseriesiitjee3.jpg

    Harmonic Progression

    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:

    http://www.jagranjosh.com/imported/images/E/Articles/sequencesandseriesiitjee2.jpg

    Relation between Arithmetic Mean & Geometric Mean

    Relation between Arithmetic Mean & Geometric Mean

    Here, we can see that the sequence 2, 6, 18 is a geometric progression.

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

    https://files.askiitians.com/cdn1/images/2017817-121042530-588-equation-7.png

    Where a and b are the positive integers.

    Example: Find two numbers, If arithmetic mean and 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.

    Special Series

    Special Series

    Special Series are the series which are 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 result:

    Practice Questions on Sequence & Series

    Practice Questions on Sequence & Series

    Question 1: Find the sum of the following series:

    http://www.hitbullseye.com/mba/quant/images/Sequences-&-Series-6.png

    Question 2: If xa = yb = zc, where a, b, c are unequal positive numbers and x, y, z are in GP, then a3 + c 3is

    1. >2b3
    2. > 2b
    3. < 2b3
    4. <2b

    Question 3: The maximum value of the sum of the AP: 99, 96, 93,....... is:

    1. 1883
    2. 2083
    3. 1683
    4. 983

    Question 4: If ai > 0, i = 1, 2, 3, ….., 50 and a1 + a2 + a3 + …….. + a50 = 50, then the minimum value of is equal to:

    1. 100
    2. 50
    3. 150
    4. 200

    Question 5: If is equal to:

    Question 6: The value of 

    1. 1
    2. 2
    3. 3/2
    4. 4

    Question 7: The first two terms of a geometric progression add up to 12. The sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is:

    1. -4
    2. -12
    3. 12
    4. 4

    Question 8: If ????, ????, ???? are in A.P., then (???? + 2???? − ????) (2???? + ???? − ????) (???? + ???? − ????) equals:

    1. 12????????????
    2. ????????????
    3. 2????????????
    4. 4????????

    Tips & Tricks to Study Sequence & Series

    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 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 by 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 of intelligent guesses. Do not rely completely on guess work.
    • Try to systematically strike down options by the process of elimination. Questions which have “All of These” as an option, this trick specially 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.

    Adoption of the above mentioned tricks and help of the study notes will give you an overview of the paper and the questions asked in the examination.

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