Harmonic Progression (HP)

Closely related to the AP is the harmonic progression.

Definition 5.2. A sequence $h_1,h_2,h_3,\ldots$ is a harmonic sequence (or harmonic progression, HP) if $\dfrac1{h_1},\dfrac1{h_2},\dfrac1{h_3},\ldots$ is an arithmetic sequence.

So a sequence is in HP exactly when its reciprocals are in AP — but one should NOT say "harmonic progressions are the reciprocals of arithmetic progressions" carelessly: if the underlying AP contains a zero term, its reciprocal is not even defined. Provided an AP $a,a+d,a+2d,\ldots$ contains no zero term (i.e. $-\tfrac ad$ is not a nonnegative integer within range), its reciprocal sequence IS a genuine HP, of the general form

Useful facts.

• $\left(\dfrac1n\right)=1,\dfrac12,\dfrac13,\ldots$ is the simplest nontrivial HP.
• If $a,b,c$ are in HP, then $b=\dfrac{2ac}{a+c}$.
• In a triangle, if the altitudes are in AP, the corresponding sides are in HP. …