Question

Insert three harmonic means between \(\frac{6}{21}\) and \(\frac{6}{5}\)

Short answer

The harmonic means between \(\frac{6}{21}\) and \(\frac{6}{5}\) are the reciprocals of \(\frac{21}{6} + \frac{-2}{3}\), \(\frac{21}{6} + 2(\frac{-2}{3})\), and \(\frac{21}{6} + 3(\frac{-2}{3})\).

Step-by-step solution

Understand Harmonic Mean

Harmonic mean between two numbers A and B for n terms is found by reversing A and B, inserting n arithmetic means between these reciprocals, and then taking the reciprocal of these terms to get the harmonic means.

Calculate the Reciprocals

First, find the reciprocals of the given numbers. The reciprocal of \(\frac{6}{21}\) is \(\frac{21}{6}\) and the reciprocal of \(\frac{6}{5}\) is \(\frac{5}{6}\).

Determine the Distance between Reciprocals

Calculate the distance (d) between the reciprocals by subtracting them: \(d = \frac{5}{6} - \frac{21}{6} = -\frac{16}{6} = -\frac{8}{3}\).

Calculate the Common Difference

To insert three means, there should be three intervals between them, therefore we have a total of 4 intervals between the original reciprocal numbers. The common difference (cd) for arithmetic means is \(cd = \frac{d}{n + 1} = \frac{-\frac{8}{3}}{3 + 1} = -\frac{2}{3}\).

Calculate the Arithmetic Means

Find the three arithmetic means (AM1, AM2, AM3) between the reciprocals: AM1 = \(\frac{21}{6} + \frac{-2}{3}\), AM2 = AM1 + \frac{-2}{3}, AM3 = AM2 + \frac{-2}{3}.

Convert Arithmetic Means to Harmonic Means

Finally, take the reciprocal of each arithmetic mean to get the harmonic means: HM1 = \(1 / \text{AM1}\), HM2 = \(1 / \text{AM2}\), HM3 = \(1 / \text{AM3}\). Calculating these will give us the harmonic means between \(\frac{6}{21}\) and \(\frac{6}{5}\).

Key concepts

Arithmetic Mean

The arithmetic mean, or simply the average, is the sum of a series of numbers divided by the count of that series of numbers. In everyday language, it's what we commonly call the 'average' of a set of values. The formula to calculate the arithmetic mean is \[\text{Arithmetic Mean} = \frac{\sum_{i=1}^{n} x_i}{n}\]where \( x_i \) represents each number in the series and \( n \) denotes the total number of terms. It is one of the most popular measures of central tendency in data. When understanding harmonic means, the concept of arithmetic mean is crucial as it is used in the process of finding harmonic means when dealing with reciprocals.

In our exercise, reciprocals of the given numbers were used to calculate arithmetic means, and then, these means were transformed back to get harmonic means. Remember, the arithmetic mean you calculate from reciprocals will not be the harmonic mean itself, but a step in the process of finding one.

Reciprocals in Mathematics

A reciprocal in mathematics is simply the inverse of a number. To find the reciprocal of a fraction, you swap the numerator and the denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \), assuming \( b eq 0 \). When two numbers are multiplied together and their product is 1, they are considered reciprocals of each other.

In the context of our exercise, finding the reciprocal is a key step before you can start inserting arithmetic means, which will later be reverted to provide the harmonic means. It's important to master finding reciprocals because they are used in many areas of mathematics, including the computation of harmonic means – which is the heart of this exercise. An interesting fact about reciprocals is that they reveal the 'harmonic' relationship between numbers, as seen in their application to calculate harmonic means.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that often follow a specific pattern or rule. These sequences can be finite or infinite, and they play a significant role in various fields of mathematics and its applications. Common types of sequences include arithmetic sequences, where each term after the first is derived by adding a constant, known as the common difference, to the preceding term. Geometric sequences, on the other hand, involve multiplying by a constant factor to move from one term to the next.

Understanding sequences is essential for our exercise as we work with an arithmetic sequence of reciprocals. The concept of inserting terms into a sequence is particularly relevant when finding harmonic means, as it requires inserting a set number of terms between two given reciprocals that follow the pattern of an arithmetic sequence. This concept provides the structure needed to systematically determine the values that complete our 'harmonically' distributed series between two given numbers.