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814 ALGEBRA. 384. The Reciprocals of Quantities in Harmonical Progression are in Arithmetical Progression.

By definition, if a, b, c are in Harmonical Progression,

a c = a - b b - c; a(b - c) - c{a - b),

dividing every term by abc,

1 c - 1 b = 1 b - 1 a

which proves the proposition. We may therefore define an Harmonical Progression as a series of quantities the reciprocals of which are in Arithmetical Progression.

385. Solution of Questions in H, P. Harmonical properties are chiefly interesting because of their importance in Geometry and in the Theory of Sound: in Algebra the proposition just proved is the only one of any importance. There is no general formula for the sum of any number of quantities in Harmonical Progression. Questions in H. P. are generally solved by inverting the terms, and making use of the properties of the corresponding A. P.

Ex. The 12th term of an H. P is 1 5, and the 19th term is 3 22: find the series.

Let a be the first term, d the common difference of the corresponding A. P. ; then

5 = the 12th term = a + 11d; and 22 3 = the 19th term = a + 18d ; whence d = 13, a = 3 2 Hence the Arithmetical Progression is 43, 53, 2, 73, ... and the Harmonical Progression is 34, 35, 12, 37, ...

386. Harmonic Mean. When three quantities are in Harmonic Progression the middle one is said to be the Harmonic Mean of the other two.