Formulas For Harmonic Progression

Formulas For Harmonic Progression

Formulas For Solving Harmonic Progression Easily

Harmonic Progression is a sequence of quantities whose reciprocals are in arithmetical progression .Each term in the Harmonic Progression is the Harmonic Mean Of its Neighbouring Term. Hence , it is  is a sequence of real numbers formed by taking the reciprocals of an arithmetic progression. mean of its two neighbors.

Definition & Formula for Harmonic Progression

  • Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression
  • Also, a sequence is a harmonic progression when each term is the harmonic mean of its neighbouring termTo solve a problem on Harmonic Progression, make the corresponding AP series and then solve the problem. For Harmonic Progression sum of integer is not possible.
  • For example: Given sequence or series is 1/2, 1/4, 1/6, 1/8, …. Take the reciprocal of each term, therefore, we get 2, 4, 6, 8,…… Now, a = 2 and d = 2

Formulas of Harmonic Progression (HP)

  • How to find nth term of an HP

Tn = 1/(a + (n – 1)d)

where tn = nth term,

a= the first term ,

d= common difference,

n = number of terms in the sequence

 

  • Harmonic Mean (HM)

Harmonic Mean is type of numerical average, which is calculated by dividing the number of observation by the reciprocal of each  number in series.

If a, b are in HP, then there HM is

HM = n / { 1/a1 + 1/a2 … 1/an}

Where, n = Total number of numbers or terms, a1, a2,…..an    =   Individual terms or individual values.

 

Properties of Harmonic Progression

  • For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then
    A≥ G ≥ H
  • Relationship between arithmetic, geometric, and harmonic means AM * HM = GM2
  • Unless a = 1 and n = 1, the sum of a harmonic series will never be an integer. This is because at least one denominator of the progression is divisible by a prime number that does not divide any other denominator.
  • Three consecutive numbers of a HP are: 1/(a–d),  1/a,  1/(a+d)
  • Four consecutive numbers of a HP are: 1/(a–3d),  1/(a–d),   1/(a+d),  1/(a+3d).

 

Read Also – How to solve Harmonic Progression Quickly

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