# 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:

${a}_{n}=\frac{1}{a+\left(n-1\right)×d}$

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:

${S}_{n}=\frac{n}{2}×\left(\frac{2a+\left(n-1\right)×d}{a+\left(n-1\right)×d}\right)$

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:

1. Reciprocal of an AP:

• In a HP, the reciprocals of the terms form an arithmetic progression.
2. Inverse Relationship:

• As opposed to AP and GP, in an HP, the terms have an inverse relationship.
3. 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.