Relation Between AP, GP and HP
Progression is also known as sequence or series, are the types of numbers that are placed in a particular order to form a recognizable table. Simply we can say that harmonic progression is the reciprocal of the values of the terms in arithmetic progression. It can be explained as if the terms of arithmetic progression like a, b, c, d, e, f are available in the form of 1/a, 1/b, 1/c, 1/d, 1/e, 1/f, in which terms of harmonic progression can be written as 1/a, 1/(a + d), 1/(a + 2d), etc. A sequence can be said in harmonic progression only when the terms are in harmonic mean with their neighboring terms.
Harmonic mean is to find out the average speed of the journey where the average formula of the speed can be given by 2ab/ (a + b) kmph, where a and b are the speed at which a person travels a distance of A and B. Harmonic mean can be easily calculated with the help of the following example;
If three terms a, b, and c are in harmonic progression then, the terms in arithmetic progression include 1/a, 1/b, 1/c,
2/b = 1/a + 1/c
b = 2ac/ (a + c)
In the above equation, b is the harmonic mean of its other two neighboring terms (a, c).
Conditions to Solve the Terms of Harmonic Progression
Mostly all the terms included in the harmonic progression are first converted into arithmetic progression and then are solved further.
To solve the problems in harmonic progression, one must make a corresponding arithmetic series in order to solve the harmonic series.
For any two given terms, if A.P, G.P, H.P are indicated as arithmetic, geometry, and harmonic progression, then
A .P ≥ G.P ≥ H.P
And, (A.P) (H.P) = (G.P)2
Through the given terms of arithmetic terms, we need to first find the first three terms of the harmonic progression and then they can be taken in its general form of 1/(a – d), 1/a, 1/(a + d).
If the nth term of the arithmetic progression is given by an = a + (n – 1) d, and we know to solve the terms in harmonic progression, we first have to convert the terms of A.P in H.P, so the nth term of the harmonic progression can be given by 1/
a+(n−1)d
a+(n−1)d.
Any terms available in the series of harmonic progression cannot be zero.
Arithmetic and Geometric Progression
Arithmetic terms are the sequence of numbers in which the difference between any two adjacent terms is constant and is also known as the common difference which is denoted by d. The common difference between any two adjacent terms can be given as follows;
d = a2 – a1 = a3 – a2 = a4 – a3 = ...
If one term and the common difference is known then we can easily calculate the value of the next terms by adding the value of the common difference in the value of the term given, which can be given as follows;
A(m+1) = am + d
The value of the nth term in an arithmetic progression can be calculated easily with the help of the following formula;
an = a1 + (n − 1) d
or, an = am + (n − m) d
The sum of the value of n terms in an arithmetic progression can be given as the n times average of the sum of the first and the last term of the arithmetic progression, which can be mathematically shown as follows;
S = (n/2) (a1 + an)
If the last term of the sequence (an) is not given but the value of the common difference is known then the sum of the n terms can be calculated easily with the help of the following formulae;
S = (n/2) [2a1 + (n − 1) d]
Geometric progression is the term in which if the first term is known and the common ratio is known then the next term of the series can be calculated easily by multiplying the first term with the fixed and non-zero term of common ratio which is denoted by r.
The common difference of any geometric progression can be easily calculated by dividing the second term with the first term and the quotient remaining is known as common difference, i.e.
r = (am+1/am) = a2/ a1 = a3/ a2 = a4/ a3 = ...
If the value of the first term and the common difference is known then we can easily calculate the value of the next term or the nth term with the help of the following formulae:
an = a1 rn-1
or, an = am rn-m
The sum of all the terms included in the sequence of geometric progression can be easily calculated with the help of the common difference and the known value of the first term in the sequence, which can be easily calculated with the help of the following formulae;
S = [a1(1 − rn)]/(1 – r)
Relation Between A.P, G.P, and H.P
The above-shown figure easily explains the relation between the different types of sequences and can be explained further in detail:
Find the values of three numbers in H.P can be easily calculated with the help of its general formulae, i.e. 1/(a – d), 1/a, 1/(a + d).
The relation between the three types of mean available in the sequence can be provided as follows;
A.M > G.M > H.M.
Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.
If a and b are the two real positive numbers given then the relation between the three types of the sequence can be given as follows;
A.M x H.M = G.M2
Where A.M is the Arithmetic Mean, G.M is the Geometric Mean and the H.M is known as Harmonic Mean.
The nth term of the arithmetic progression can be easily calculated by an = a + (n - 1) d, while the nth term of the harmonic progression can be easily calculated by 1/
a+(n−1)d
a+(n−1)d.
The most basic relation between H.P and A.P is that most of the H.P terms are calculated by first converting them into the terms of A.P.
From the above information we can conclude that the nth term of the Harmonic Progression can be calculated with the following formulae;
The nth term of the Harmonic Progression = 1/ (nth term of the corresponding Arithmetic Progression).
To check whether the sequence is in harmonic progression or not we must check if the reciprocal of the given sequence has the same difference between their consecutive terms and can be called as an arithmetic sequence then the given sequence whose
reciprocal was done can be said in harmonic progression.
Some examples are shown below to explain the concept of harmonic progression:-
Example 1
If the 7th and the 12th term of an H.P are given as 1/10 and 1/25 respectively, then to find the 20th and the nth term the following solution can be taken into consideration.
Solution:
The solution of the above example can be given with the help of the general form of the Harmonic progression.
The general form of Harmonic progression is;
1/a + 1/(a + d) + 1/(a + 2d) + ….
The value of the 7th term of the sequence is given as 1/10.
And according to the general formula, the 7th term can be given as 1/(a + 6d)
So by combining the above two equations for the 7th term, we get a + 6d = 10 …… (1)
Similarly, for the 12th term
12th term = 1/25 and 1/(a + 11d)
So after combining we get a + 11d = 25 ...... (2)
From the above two equations ( 1 and 2 ) we get,
A = -8 and d = 3 ……. (3)
Through these values of a and d, we can easily calculate the values of 20th as well as the nth term of the harmonic progression.
For the 20th term, the general form is 1/ (a + 19d), and putting the values of equation (3) in this form we get,
20th term = 1/
−8+(19∗3)
−8+(19∗3)
20th term = 1/ 49
Similarly, we can easily calculate the nth term of the harmonic progression with the help of its general form and the values of a and d given in equation (3).
For the nth term, the general form is given as 1/
a+(n–1)d
a+(n–1)d and by putting the values of a and d from equation (3) in this form we can easily calculate the value of the nth term of the harmonic sequence.
nth term = 1/
a+(n–1)d
a+(n–1)d
nth term = 1/
−8+(n–1)3
−8+(n–1)3
nth term = 1/ (3n – 11)
Hence, the values of the 20th and the nth term are easily found out in the above example with the help of the two terms of the harmonic progression provided.
Important Facts to Remember
The sequence of real numbers that is determined by noting the reciprocals of the arithmetic progression which does not contain zero is known as Harmonic progression.
A term in the sequence of harmonic progression is considered to be the harmonic means of its neighbors.
In order to solve the harmonic progression problems, students should perceive the corresponding sum of arithmetic progression.
The nth term of the harmonic progression must be equal to the nth terms reciprocal of the corresponding Arithmetic Progression.
The four kinds of sequences are Harmonic sequences, Arithmetic sequences, Geometric sequences, and Fibonacci numbers.
In order to find the common difference, two- adjacent terms have to be subtracted. Every term can be found by recursively adding d, i.e. the common difference between the preceding term.
The calculating harmonic mean is important because it helps in perceiving the multiplicative and divisor relationships between the given fractions without thinking about the common denominators.
There are several uses of Harmonic sequences that can be applied in everyday life.
Harmonic formulae can be used by scientists for concluding their experiments.
Harmonic progression establishes the way water boils when the temperature is changed with the same values.
The harmonic sequence is also used in the musical notes.
The number of raindrops is determined by harmonic progression by calculating the amount of rainfall by creating an illusion that it can be calculated when the series is infinite.
Traffic bunches are real examples of harmonic sequence.
Quick Ways to Master this Chapter Like an Expert
Remember The Formulae
Students must pay closer attention to the formulas of Harmonic progression in order to solve the sums more effectively and within less time. Calculating the series and sequences can be time-consuming so an aspirant must be well versed with the formulas and relationships between the various sequences.
Practice regularly
Math is a subject that requires constant practice and so does this chapter. In order to solve the sums like an expert, an aspirant must practice the sums regularly to stay in touch with the concepts and formulas of harmonic progression and its series and sequences.
Solve questions from RD Sharma textbook.
An aspirant should solve the questions from the textbook in order to prepare for the exam. Practicing sums of Harmonic progression from RD Sharma can give an idea of possible questions to the aspirants who are preparing for JEE Advanced.
Practice solved examples
The solved examples establish the fundamental knowledge and idea about AP, GP, and HP for the aspirants. It can be helpful in understanding the basic concepts of the chapter including the formulas for series and sequences.
Attend classes on Harmonic progression
Paying attention to the teacher and the classes can help an aspirant understand the chapter in the best possible way. An aspirant must be willing to listen and learn carefully about Harmonic progression in order to solve the sums and calculate the integers and consecutive terms of HP.
Make notes during class
An aspirant must make notes simultaneously during the classes so that the notes can help later during revision sessions. It is very important to note the formulas and rules in order to calculate the sums correctly.
Stay in touch with Harmonic Progression with the help of sample papers
An aspirant must be willing to solve the sample papers after solving the examples and questions from the textbook in order to truly master the chapter. Harmonic progression can have various possible questions for exams so it is always better to be prepared.
Test knowledge with quizzes
After solving questions from the textbooks, aspirants may test their knowledge through quizzes which will help them clock their time taken for solving one sum. It is a great way to challenge their skills on a lighter note.
Participate in online interactive classes
Aspirants can participate in online classes in order to clear their doubts as well as for the revision of this chapter. It will surely help the students understand Harmonic Progression in more efficient ways. It will give them room to make notes and answer the questions to test their skills.
JEE aspirants can access the online materials such as notes, practice papers, sample questions, solutions to RD Sharma book along with puzzles and quizzes on Harmonic Progression from Vedantu’s official website. The solutions can be downloaded in PDF formats as well and if an aspirant is stuck at any point in Harmonic Progression then the aspirants can access the online live classes and doubt clearing sessions in order to progress in the chapter.
