Formulas For Harmonic Progression

Definition & Formula for Harmonic Progression

Harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression.

NOTE: To 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 for Harmonic Progression

Formulas of Harmonic Progression (HP)

  • 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 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)

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