Formulas For Harmonic Progression

Harmonic Progression Formulas in Aptitude

Harmonic Progression is a Sequence of quantities whose reciprocals are in Arithmetical progression. We have Limited Harmonic progression Formulas .Each term in the Harmonic Progression is the Harmonic Mean of its Neighboring Term. Hence, it is is a sequence of real numbers formed by taking the reciprocals of an Arithmetic progression. mean of its two neighbors.

HP is Represented in the form of \mathbf{\frac{1}{a_{1}} , \frac{1}{a_{2}} ,\frac{1}{a_{3}} ……\frac{1}{a_{n}}}

In this Page Harmonic Progression Formulas is given that is useful to Solve many Problems in different Competitive Examinations.

Harmonic Progression Formulas

Definition of Harmonic Progression

  • Harmonic progression is a progression formed by taking the reciprocals of an Arithmetic progression.
  • A sequence is a harmonic progression when each term is the harmonic mean of its Neighboring term.
  • 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 \mathbf {\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{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

a_{n} = \mathbf{\frac{1}{(a + (n – 1)d)}}

where a_{n} = nth term,

\frac{1}{a}= the first term ,

d= common difference taken from AP,

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 = \mathbf{\frac{n} {\frac{1}{a_{1}} + \frac{1}{a_{2}} ……\frac{1}{a_{n}}}}

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

          Harmonic Mean of two numbers a and b is \mathbf{\frac{2ab}{a+b}}

          Harmonic Mean of three numbers a, b and c is \mathbf{\frac{3abc}{ab+bc+ca}}

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 \times  HM = GM^{2}
  • 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: \mathbf{\frac{1}{(a–d)},  \frac{1}{a},  \frac{1}{(a+d)}}
  • Four consecutive numbers of a HP are: \mathbf{\frac{1}{(a–3d)},  \frac{1}{(a–d)},   \frac{1}{(a+d)},  \frac{1}{(a+3d)}}.

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