Exercise 4.4 — Question 2: Harmonic Sequences and Series

Key Definitions

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 $d$.

The general (nth) term of a harmonic sequence is: $a_n=\frac{1}{a+(n-1)d}$ where $a=\frac{1}{a_1}$ is the first term of the corresponding arithmetic sequence and $d$ is the common difference of that arithmetic sequence.

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Worked Example

Problem: Find the $n$th term and the sum of the first $n$ terms of the harmonic series whose first two terms are $\frac{1}{2}$ and $\frac{1}{4}$.

Step 1 — Identify the corresponding arithmetic sequence.

Take reciprocals of the harmonic terms: $2,4,\dots$ This is an arithmetic sequence with first term $a=2$ and common difference $d=4-2=2$.

Step 2 — Write the general term of the arithmetic sequence. $A_n=a+(n-1)d=2+(n-1)(2)=2n$

Step 3 — Write the general term of the harmonic sequence. $a_n=\frac{1}{A_n}=\frac{1}{2n}$

Step 4 — Write the harmonic series. $S_n=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots +\frac{1}{2n}=\frac{1}{2}\left(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}\right)$

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Harmonic Mean

The harmonic mean $H$ of two numbers $a$ and $b$ is: $H=\frac{2ab}{a+b}$

If $H$ is the harmonic mean between $a$ and $b$, then $a,H,b$ form a harmonic sequence, which means $\frac{1}{a},\frac{1}{H},\frac{1}{b}$ form an arithmetic sequence.

Verification: $\frac{1}{H}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H}$, confirming equal differences.

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Key Properties