4.3 Q-3

Exercise 4.3 — Question 3

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 with some common difference $d$.

General term of a harmonic sequence: $a_n=\frac{1}{\frac{1}{a_1}+(n-1)d}$

where $\frac{1}{a_1}$ is the first term of the corresponding arithmetic sequence and $d$ is the common difference of that arithmetic sequence.

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Key Relationships

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Worked Example

Problem: If the 3rd term of a harmonic sequence is $\frac{1}{7}$ and the 7th term is $\frac{1}{15}$, find the general term and the 10th term.

Step 1: Convert to the corresponding arithmetic sequence.

Let $b_n=\frac{1}{a_n}$ be the corresponding arithmetic sequence.

$b_3=7\text{and}{b}_7=15$

Step 2: Use the arithmetic sequence formula $b_n=b_1+(n-1)d$.

$b_3=b_1+2d=7\cdots (1)$ $b_7=b_1+6d=15\cdots (2)$

Step 3: Subtract equation (1) from (2):

$4d=8⟹d=2$

Substitute back into (1): $b_1+2(2)=7⟹b_1=3$

Step 4: General term of the arithmetic sequence: $b_n=3+(n-1)(2)=2n+1$

Step 5: General term of the harmonic sequence: $a_n=\frac{1}{b_n}=\frac{1}{2n+1}$

Step 6: Find the 10th term: $a_{10}=\frac{1}{2(10)+1}=\frac{1}{21}$

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Harmonic Mean

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

This is because $\frac{1}{a},\frac{1}{H},\frac{1}{b}$ must form an arithmetic sequence, so: $\frac{1}{H}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H}⟹\frac{2}{H}=\frac{1}{a}+\frac{1}{b}$

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Strategy for Solving Harmonic Sequence Problems

1. Take reciprocals of all terms to get the corresponding arithmetic sequence.
2. Apply arithmetic sequence formulas to find unknowns.
3. Convert back by taking reciprocals to get harmonic sequence terms.