What are harmonic sequences

A sequence is a list of elements/objects arranged in order.

Some of the most common sequences are:

• Arithmetic Sequence

• Geometric Sequence

• Harmonic Sequence

In this article, we will discuss the harmonic sequence.

A series of numbers is in harmonic sequence if the reciprocals of all the sequence elements form an arithmetic sequence, which does not contain $$0$$.

In a harmonic sequence, any terms in the sequence are considered the harmonic means of its two neighbours.

Example: The sequence $$a, b, c, \cdots$$ is considered an arithmetic sequence. Then the harmonic sequence can be written as follows:

$$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \cdots$$

E2.7A: Continue a given number sequence

A number sequence is an ordered list controlled by a pattern or rule. The number in the sequence is called terms. A sequence is finite if it consists of a limited number and infinite if it consists of unlimited numbers.

Harmonic graph is used to plot the harmonic motions or the harmonic series like $$\frac{1}{3}, \frac{1}{6}, \frac{1}{9}, \cdots$$.

Worked example of giving the next number in a harmonic sequence

Example 1: Determine if the below sequence is a harmonic sequence or not. And sketch the graph.

$$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \cdots$$

Step 1: Write the given sequence.

$$\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \cdots$$

Step 2: Write the arithmetic sequence of the harmonic sequence.

$$2, 3, 4, 5, \cdots$$

Step 3: Determine whether the above sequence is an arithmetic sequence or not. 

The given is an arithmetic sequence because it is possible to obtain the preceding number. The consecutive number is obtained by adding $$2$$ to the preceding integer.

So, the given sequence is a harmonic sequence.

Step 4: Draw the graph of the given harmonic sequence.

E2.7B: Recognise patterns in a sequence, including the term-to-term rule and relationships between different sequences.

The general difference means that the difference between two consecutive numbers in a row is the same. It is the common difference denoted as $$d$$.

Worked example of recognising patterns in a harmonic sequence

Example 1: What is the common difference in the below-given order?

$$\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \cdots$$

Step 1: Write the given sequence.

$$\frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \cdots$$

Step 2: Recall the rule of the harmonic sequence.

A series of numbers is in harmonic sequence if the reciprocals of all the sequence elements form an arithmetic sequence, which does not contain $$0$$.

Step 3: Write the given sequence in arithmetical sequence.

$$5, 10, 15, 20, \cdots$$

Step 4: Write the difference of the numbers.

$$10-5=5$$

$$15-10=5$$

$$20-15=5$$

Step 5: Write the common difference.

The common difference of the arithmetic sequence is $$5$$.

Step 6: Use the rule of the harmonic sequence to write the common difference of the sequence.

The common difference of the given harmonic sequence is $$\frac{1}{5}$$.

E2.7C: Find and use the $$n\text{th}$$ term of a sequence.

The term at the $$n\text{th}$$ place of a harmonic progression is the reciprocal of the $$n\text{th}$$ term in the corresponding arithmetic progression. The following formula mathematically represents this:

The $$n\text{th}$$ term of a harmonic sequence is $$\frac{1}{\left(a+(n-1)d \right )}$$, where $$a$$ is the first term in the arithmetical sequence, $$d$$ is the common difference and $$n$$ is the number of terms in the arithmetic sequence. It can also be written as follows:

$$\text{The nth term of Harmonic Sequence}=\frac{1}{\text{nth term of corresponding Arithmetic Sequence}}$$

Worked example of finding the $$n\text{th}$$ term

Example 1: Compute the $$100$$th term of a harmonic sequence if the $$10$$th and $$20$$th term of the sequence are $$20$$ and $$40$$, respectively.

Step 1: Recall the formula for the $$n\text{th}$$ term of the arithmetic sequence.

Arithmetic sequence formula is $$a_{n}=a_{1}+(n-1)d$$

Step 2: Write the corresponding arithmetic sequence to the given harmonic sequence.

The $$10$$th term of the arithmetic sequence is as follows:

$$a+9d=\frac{1}{20}$$ 

The $$20$$th term of the arithmetic sequence is as follows:

$$a+19d=\frac{1}{40}$$

Step 3: Solve equations $$a+9d=\frac{1}{20}$$ and $$a+19d=\frac{1}{40}$$

Subtract equations (1)-(2).

$$a+9d-a-19d=\frac{1}{20}-\frac{1}{40}$$

$$-10d=\frac{1}{40}$$

$$d=\frac{-1}{400}$$

Step 4: Substitute $$d=\frac{-1}{400}$$ in $$a+9d=\frac{1}{20}$$

$$a+9(\frac{1}{400})=\frac{1}{20}$$

$$a=\frac{1}{20}+\frac{9}{400}$$

$$a=\frac{20}{400}+\frac{9}{400}$$

$$a=\frac{29}{400}$$

Step 5: Write the $$100$$th term.

$$a+99d=\frac{29}{400}+99\frac{-1}{400}$$

$$a+99d=\frac{29}{400}-\frac{99}{400}$$

a+99d=\frac{29-99}{400}$$

a+99d=\frac{-70}{400}$$

$$a+99d=\frac{-7}{40}$$

$$\text{The 100th term of the Harmonic Sequence}=\frac{1}{\text{100th term of the Arithmetic Sequence}}$$.

Therefore, the $$100$$th term of the harmonic sequence is $$\frac{-40}{7}$$.