Exercise 4.5 — Question 5: Harmonic Sequences and Series

Key Concepts

A harmonic sequence is a sequence whose reciprocals form an arithmetic sequence.

If $a_1,a_2,a_3,\dots$ is an arithmetic sequence with first term $a$ and common difference $d$, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ is a harmonic sequence.

General Term of a Harmonic Sequence

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

Harmonic Mean

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

This is derived from the fact that $\frac{1}{a},\frac{1}{H},\frac{1}{b}$ must be in arithmetic progression.

Worked Example

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

Solution:

The reciprocals are $3,7,11,\dots$, which form an arithmetic sequence with:

• First term $a=3$
• Common difference $d=4$

The $n$-th term of the arithmetic sequence is: $a_n=3+(n-1)(4)=4n-1$

Therefore, the $n$-th term of the harmonic sequence is: $T_n=\frac{1}{4n-1}$

Strategy for Solving Harmonic Problems

1. Take reciprocals of the harmonic sequence terms.
2. Verify the reciprocals form an arithmetic sequence (constant difference).
3. Use arithmetic sequence formulas on the reciprocals.
4. Convert back by taking reciprocals of the result.