4.4 Q-6

Exercise 4.4 — Question 6

Topic: Harmonic Sequences and Series

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence. If $a_1,a_2,a_3,\dots$ is a harmonic sequence, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ is an arithmetic sequence.

Key Formulas

General term of a Harmonic Sequence: $a_n=\frac{1}{a+(n-1)d}$ where $a=\frac{1}{a_1}$ is the first term of the corresponding AP and $d$ is the common difference of the corresponding AP.

Harmonic Mean between two numbers $a$ and $b$: $H=\frac{2ab}{a+b}$

Worked Example (Exercise 4.4, Q-6)

To solve problems involving harmonic sequences:

1. Take reciprocals of the harmonic sequence terms to obtain an arithmetic sequence.
2. Apply AP formulas to the reciprocals.
3. Take reciprocals again to return to the harmonic sequence.

Example: Find the $n$th term of the harmonic sequence $\frac{1}{3},\frac{1}{7},\frac{1}{11},\dots$

Solution:

• Reciprocals form AP: $3,7,11,\dots$
• First term $a=3$, common difference $d=4$
• $n$th term of AP: $a_n=3+(n-1)(4)=4n-1$
• $n$th term of HP: $T_n=\frac{1}{4n-1}$

Example: Insert 3 harmonic means between $\frac{1}{2}$ and $\frac{1}{10}$.

Solution:

• Reciprocals: insert 3 arithmetic means between $2$ and $10$
• AP: $2,\_,\_,\_,10$ with $n=5$ terms
• $d=\frac{10-2}{5-1}=2$
• AP terms: $2,4,6,8,10$
• Harmonic means: $\frac{1}{4},\frac{1}{6},\frac{1}{8}$

Summary Table