4.6 Harmonic Progression

Definition

A Harmonic Progression (HP) is a sequence of numbers whose reciprocals form an Arithmetic Progression (AP).

If $a_1,a_2,a_3,\dots ,a_n$ is a harmonic progression, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots ,\frac{1}{a_n}$ is an arithmetic progression.

Example: $1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ is an HP because $1,2,3,4,\dots$ is an AP.

---

General Term (nth Term) of an HP

Since the reciprocals of an HP form an AP, let the corresponding AP have first term $a$ and common difference $d$.

The $n$th term of the AP is: $T_n^{\text{AP}}=a+(n-1)d$

Therefore, the $n$th term of the HP is: $T_n=\frac{1}{a+(n-1)d}$

Example: Find the 8th term of the HP $1,\frac{1}{3},\frac{1}{5},\dots$

The reciprocals form the AP: $1,3,5,\dots$ with $a=1$, $d=2$.

$T_8^{\text{AP}}=1+(8-1)(2)=1+14=15$

$T_8^{\text{HP}}=\frac{1}{15}$

---

Harmonic Mean

The Harmonic Mean (HM) of two numbers $a$ and $b$ is: $H=\frac{2ab}{a+b}$

This is derived from the condition that $a,H,b$ are in HP, which means $\frac{1}{a},\frac{1}{H},\frac{1}{b}$ are in AP: $\frac{1}{H}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H}⟹\frac{2}{H}=\frac{1}{a}+\frac{1}{b}=\frac{a+b}{ab}$ $\therefore H=\frac{2ab}{a+b}$

Example: Find the harmonic mean between $3$ and $6$. $H=\frac{2(3)(6)}{3+6}=\frac{36}{9}=4$

---

Inserting $n$ Harmonic Means Between Two Numbers

To insert $n$ harmonic means between $a$ and $b$:

1. Take reciprocals: insert $n$ arithmetic means between $\frac{1}{a}$ and $\frac{1}{b}$.
2. The common difference of the resulting AP is: $d=\frac{\frac{1}{b}-\frac{1}{a}}{n+1}=\frac{a-b}{ab(n+1)}$
3. The $k$th harmonic mean ($k=1,2,\dots ,n$) is: $H_k=\frac{1}{\frac{1}{a}+k\cdot d}$

Example: Insert 2 harmonic means between $\frac{1}{2}$ and $\frac{1}{8}$.

Reciprocals: insert 2 arithmetic means between $2$ and $8$. $d=\frac{8-2}{2+1}=\frac{6}{3}=2$ AP: $2,4,6,8$ → HP: $\frac{1}{2},\frac{1}{4},\frac{1}{6},\frac{1}{8}$

The two harmonic means are $\frac{1}{4}$ and $\frac{1}{6}$.

---

Relationship Between AM, GM, and HM

For two positive numbers $a$ and $b$:

• Arithmetic Mean: $A=\frac{a+b}{2}$
• Geometric Mean: $G=\sqrt{ab}$
• Harmonic Mean: $H=\frac{2ab}{a+b}$

Key relationship: $A\ge G\ge H\text{(for positive }a,b\text{)}$

Also: $G^2=A\cdot H$, i.e., $G$ is the geometric mean of $A$ and $H$.

Proof: $A\cdot H=\frac{a+b}{2}\cdot \frac{2ab}{a+b}=ab=G^2$ ✓

---

Key Points to Remember