Harmonic progressions and means

Maths · Sequences and series. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A sequence is a harmonic progression (H.P.) if the reciprocals of its terms are in arithmetic progression. For example 1/3, 1/5, 1/7, ... is an H.P. because 3, 5, 7, ... is an A.P.

There is no separate set of formulae to learn for the terms. Turn every term upside down, work with the A.P., then turn the answer upside down again. A harmonic mean (H.M.) between two numbers is found in the same way.

For two numbers a and b the three means A, G and H are linked. If a and b are positive and different and G is taken as the positive root, A is the largest and H is the smallest. G² = AH always holds.

Rules to remember

a₁, a₂, a₃, ... are in H.P. if 1/a₁, 1/a₂, 1/a₃, ... are in A.P.
nth term of an H.P. = 1 ÷ (nth term of the matching A.P.) = 1/(a + (n − 1)d)
H.M. between a and b: H = 2ab/(a + b)
n harmonic means between a and b: find n arithmetic means between 1/a and 1/b, then take reciprocals
A = (a + b)/2, G = ±√(ab) and H = 2ab/(a + b)
G² = AH
A > G > H for two different positive numbers, taking G = +√(ab) (A = G = H if a = b)

Common mistake

Students add a common difference to the terms of the H.P. itself. The terms of an H.P. do not have a common difference: only their reciprocals do, so always change to the A.P. first.

Worked example

Find the harmonic mean H between 4 and 12 and check that G² = AH.

H = 2ab/(a + b) = (2 × 4 × 12)/(4 + 12) = 96/16 = 6.
A = (4 + 12)/2 = 8 and G² = ab = 4 × 12 = 48.
AH = 8 × 6 = 48, which equals G².

Answer: H = 6, and G² = AH = 48

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