Harmonic Sequence: Definition and Examples

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Harmonic Sequence: Definition and Examples

A harmonic sequence is a sequence of numbers in which the reciprocals of the terms are in arithmetic progression. In other words, a sequence is called a harmonic sequence if the difference between the reciprocals of consecutive terms is constant.

The general form of a harmonic sequence can be written as:

a, \frac{1}{a+d}, \frac{1}{a+2 d}, \frac{1}{a+3 d}, \text { dots }

where a is the first term and d is the common difference between the reciprocals of consecutive terms.

To understand the concept better, let’s go through a few examples.

Example 1: Consider the sequence 1,

1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \backslash \text { dots. }

In this sequence, the first term a is 1, and the common difference between the reciprocals of consecutive terms is-\frac{1}{n}, where n represents the position of each term in the sequence.

Therefore, the harmonic sequence can be written as:

1, \frac{1}{1+1}, \frac{1}{1+2}, \frac{1}{1+3}, \frac{1}{1+4}, \text { dots }

Example 2:

Consider the sequence 2,

2, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \frac{1}{10}, \text { dots. }

In this sequence, the first term a is 2, and the common difference between the reciprocals of consecutive terms is -\frac{1}{2 n}.

Therefore, the harmonic sequence can be written as:

2, \frac{1}{2+1}, \frac{1}{2+2}, \frac{1}{2+3}, \frac{1}{2+4}, \text { dots }

In both examples, the reciprocals of consecutive terms form an arithmetic progression, making the sequences harmonic.

Now that we understand what a harmonic sequence is, let’s move on to understanding its properties and finding the sum of an infinite harmonic series.

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