# Harmonic sequences

**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 $$\frac{-40}{7}$$.