Harmonic Progressions: Concept & Tricks

Introduction
A series of terms is known as a HP series when the reciprocals of elements are in arithmetic progression. E.g.,1/a, 1/(a+d), 1/(a + 2d), and so on are in HP as a, a + d, a + 2d are in AP. In other words, the inverse of a harmonic sequence follows the rule of an arithmetic progression.
Example of harmonic progression is 1/2, 1/4, 1/6, ...
If we take the reciprocal of each term of the above HP, the sequence will become 2, 4, 6, …. which is an AP with common difference of 2.
Harmonic Progression Formula: The general form of a harmonic progression:

hello 1
  • The nth term of a Harmonic series is:

    hello 1
  • In order to solve a problem on Harmonic Progression, one should make the corresponding AP series and then solve the problem.
Harmonic Mean: If three terms a, b, c are in HP, then 1/a, 1/b and 1/c form an A.P.
Therefore, harmonic mean formula-
2/b = 1/a + 1/c
The harmonic mean b = 2ac/(a + c)
Important points:
  • 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 harmonic progression are: 1/(a–d), 1/a, 1/(a+d)
  • Four consecutive numbers of a harmonic series are: 1/(a–3d), 1/(a–d), 1/(a+d), 1/(a+3d)
  • Five consecutive numbers of a harmonic progression are: 1/(a–2d), 1/(a–d), 1/a, 1/(a+d), 1/(a+2d)
  • If a1, a2, ……, an are n non-zero numbers, then the harmonic mean H of these numbers is given by 1/H = 1/n (1/a1 + 1/a2 +...+ 1/an).