4.3 Q-5

Exercise 4.3 — Question 5: Harmonic Sequences

Key Concepts

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 with common difference $d$.

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 arithmetic sequence and $d$ is the common difference.

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

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Typical Problem Type

Problem: Find the $n$th term of the harmonic sequence whose first two terms are given, e.g., $\frac{1}{3},\frac{1}{7},\frac{1}{11},\dots$

Step-by-step solution:

1. Take reciprocals to get the arithmetic sequence: $3,7,11,\dots$
2. Identify $a_1=3$ and common difference $d=7-3=4$.
3. General term of the A.P.: $T_n=3+(n-1)(4)=4n-1$
4. Therefore, the $n$th term of the harmonic sequence is: $a_n=\frac{1}{4n-1}$

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Another Common Problem Type

Problem: Insert $n$ harmonic means between two numbers $a$ and $b$.

Method:

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

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Important Notes

• There is no simple closed-form formula for the sum of $n$ terms of a harmonic series; each problem is solved by converting to the corresponding A.P.
• The harmonic mean $H$ of $a$ and $b$ satisfies: $\frac{1}{H}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)$
• For positive numbers: $A.M.\ge G.M.\ge H.M.$