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Numbers are sorted in a predictable order when they are placed in a specific sequence. The nth term of a progression is computed using a specific formula, whereas the nth term of a sequence is computed using certain logical criteria. Arithmetic progression, Geometric progression, and Harmonic progression are the three types of progression.
In harmonic progression, any phrase in the series is treated as the harmonic mean of its two preceding terms. Learn everything there is to know about harmonic progression formula meaning and other details.
What is Harmonic Progression?
A harmonic progression is a set of real numbers that may be determined by calculating the reciprocals of an arithmetic progression that doesn’t contain any negative integers. Any word in a series is regarded as the harmonic mean.
For example, if the sequence (a, b, c, d….) is depicted as an arithmetic progression, then the harmonic progression would be:
(1/a,1/b,1/c,1/d…….)
Harmonic Progression Formulas
The harmonic progression formulas can be used in a variety of computations.
The nth term of a harmonic progression: It’s the reciprocal of the arithmetic progression’s nth term. The reciprocal of the sum of the first term and the (n – 1) times of the common difference is the nth term of the harmonic progression. The nth term can be used to locate any of the harmonic sequence’s terms.
nth Term of HP = 1/(a + (n – 1)d), where HP is harmonic progression
Examples of Harmonic Progression
The following are some examples of harmonic progression formula:
- (½,⅓,¼,⅕,…..)
2.( ⅕,1/10,1/15,1/20,…..)
- (¼,1/7,1/11….).
Harmonic Progression Sum
If (1/a, 1/a + d,1/a + 2d,……… , 1/a + (n – 1)d) a harmonic progression is given, the formula to compute the sum of n terms in the harmonic progression is:
Sum of n terms,Sn =(1)/(d)ln [(2a +(2n – 1) d)(2a – d)]
where,
“(a)” refers to the first term of A.P
“(d)” refers to the common difference of A.P
“(ln)” refers to the natural logarithm
Relationship Between Arithmetic Mean, Geometric Mean, and Harmonic Mean
For a given set of arithmetic mean, geometric mean, and harmonic mean, the arithmetic mean is greater, followed by the geometric mean, and finally, the harmonic mean.
Arithmetic mean > geometric mean > harmonic mean
The product of the arithmetic and harmonic means equals the square of the geometric mean.
Geometric mean2 = Arithmetic mean ✕ Harmonic mean
Applications of Harmonic Progression
In addition to math, engineering, physics, and business, harmonic sequence and harmonic mean have a wide range of applications. The following are some of the most important harmonic progression formula applications.
- Scientists can employ harmonic progression formulas to determine the worth of their experiments in everyday life.
- Harmonic progression is a method of determining how water boils when the temperature is adjusted by the same amount each time.
- Harmonic sequencing is a notion that is employed in music.
- The number of droplets is counted using harmonic progression; calculating the amount of rainfall produces the illusion that it may be approximated when the series is endless.
- Traffic jams – this is another example of a harmonic sequence in action.
- In geometry, the radius of a triangle’s incircle is one-third of the harmonic mean of the triangle’s altitudes.
- The profit earnings ratio is calculated in the realm of finance utilising the notion of the weighted harmonic mean of separate components.
Facts about Harmonic Progression
- To solve a problem using a harmonic progression, first create the arithmetic progression series and then solve the problem.
- As the nth term of an arithmetic progression is given by an is equal to a + (n-1) d, so it can be said that the nth term of an harmonic progression is given by 1/ [a + (n -1) d].
- If the arithmetic, geometric, and harmonic means of two numbers are A, G, and H, respectively, then
- A ≥ G ≥ H
- A H = G2, i.e., A, G, H are in GP
- If three numbers are needed in harmonic progression, they should be provided as 1/a–d, 1/a, 1/a+d.
- The vast majority of harmonic progression questions are answered by first translating them into arithmetic progression.
Conclusion
That’s a wrap to the harmonic progression formula meaning, a harmonic progression is a series of real-number discoveries made of an arithmetic progression that does not contain (0). The harmonic progression formula can be used in a variety of situations in our daily lives. Understanding how water boils when the temperature increases by the same amount is aided by harmonic progression. It is also used to count the number of raindrops that fall.
