Harmonic Progression
Definition:
A harmonic progression (HP) is a sequence of numbers in which the reciprocals of the terms form an arithmetic progression.
General Form:
A HP can be represented as: $\frac{1}{a},\frac{1}{a+d},\frac{1}{a+2d},\dots $
 $a$ is the first term of the HP.
 $d$ is the common difference between the reciprocals of the terms.
nth Term of an HP:
The nth term of an HP is given by the formula:
${a}_{n}=\frac{1}{a+(n1)\times d}$
Where:
 ${a}_{n}$ is the nth term of the HP.
 $a$ is the first term of the HP.
 $d$ is the common difference between the reciprocals of the terms.
 $n$ is the term number.
Sum of First $n$ Terms of an HP:
The sum of the first $n$ terms of an HP (${S}_{n}$) can be calculated using the formula:
${S}_{n}=\frac{n}{2}\times \left(\frac{2a+(n1)\times d}{a+(n1)\times d}\right)$
Where:
 ${S}_{n}$ is the sum of the first $n$ terms.
 $n$ is the number of terms.
 $a$ is the first term.
 $d$ is the common difference.
Properties of Harmonic Progression:

Reciprocal of an AP:
 In a HP, the reciprocals of the terms form an arithmetic progression.

Inverse Relationship:
 As opposed to AP and GP, in an HP, the terms have an inverse relationship.

Applications:
 Used in physics for harmonic oscillations, wave mechanics, etc.
 In finance for average rates, ratios, etc.
Applications of HP:
 Physics:
 Models oscillatory systems, sound waves, pendulum motion, etc.
 Finance:
 Deals with concepts like average rates and ratios.