Understanding Harmonic Progression and its Relationship to other Progressions

The question states that three terms, \((a + b)\), \(2b\), and \((b + c)\), are in Harmonic Progression (HP). We need to determine the correct relationship between \(a\), \(b\), and \(c\) based on this information.

What is a Harmonic Progression?

A sequence of non-zero numbers is said to be in Harmonic Progression (HP) if the reciprocals of the numbers are in Arithmetic Progression (AP). For example, if \(x, y, z\) are in HP, then \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in AP.

Applying the HP Definition to the Given Terms

Given that \((a + b)\), \(2b\), and \((b + c)\) are in HP, their reciprocals must be in AP. The reciprocals are:

• \(\frac{1}{a + b}\)
• \(\frac{1}{2b}\)
• \(\frac{1}{b + c}\)

Since these three terms are in AP, the middle term is the average of the first and third terms. This can be expressed as twice the middle term being equal to the sum of the first and third terms:

\(2 \times \left(\frac{1}{2b}\right) = \frac{1}{a + b} + \frac{1}{b + c}\)

Solving the Equation

Let's simplify and solve the equation:

\(\frac{2}{2b} = \frac{1}{a + b} + \frac{1}{b + c}\)

\(\frac{1}{b} = \frac{(b + c) + (a + b)}{(a + b)(b + c)}\)

\(\frac{1}{b} = \frac{a + 2b + c}{(a + b)(b + c)}\)

Now, cross-multiply:

\(1 \times (a + b)(b + c) = b \times (a + 2b + c)\)

Expand both sides:

\(ab + ac + b^2 + bc = ab + 2b^2 + bc\)

Subtract \(ab\) and \(bc\) from both sides of the equation:

\(ac + b^2 = 2b^2\)

Subtract \(b^2\) from both sides:

\(ac = 2b^2 - b^2\)

\(ac = b^2\)

Interpreting the Result \(b^2 = ac\)

The relationship \(b^2 = ac\) is the characteristic property of a Geometric Progression (GP). If three non-zero numbers \(a, b, c\) are in GP, then the ratio of consecutive terms is constant, i.e., \(\frac{b}{a} = \frac{c}{b}\). Cross-multiplying this ratio gives \(b \times b = a \times c\), or \(b^2 = ac\).

Evaluating the Options

Let's check the given options based on our finding that \(b^2 = ac\).

1. \(a, b\) and \(c\) are in AP: This means \(2b = a + c\). This is not the same as \(b^2 = ac\). So, this option is incorrect.
2. \(a - b, b - c\) and \(c - a\) are in AP: This means \(2(b - c) = (a - b) + (c - a)\). Simplifying this gives \(2b - 2c = a - b + c - a\), which is \(2b - 2c = c - b\), leading to \(3b = 3c\), or \(b = c\). This condition (\(b=c\)) is not necessarily true if \(b^2 = ac\) unless \(a=b=c\). So, this option is incorrect.
3. \(a, b\) and \(c\) are in GP: This means \(\frac{b}{a} = \frac{c}{b}\), or \(b^2 = ac\). This matches our derived relationship. So, this option is correct.
4. \(a - b, b - c\) and \(c - a\) are in GP: This means \((b - c)^2 = (a - b)(c - a)\). Expanding this gives \(b^2 - 2bc + c^2 = ac - a^2 - bc + ab\). Substituting \(b^2 = ac\) gives \(ac - 2bc + c^2 = ac - a^2 - bc + ab\). Subtracting \(ac\) from both sides gives \(-2bc + c^2 = -a^2 - bc + ab\), which simplifies to \(a^2 + c^2 - bc - ab = 0\). This is not necessarily true if \(b^2 = ac\). So, this option is incorrect.

Based on the analysis, the correct relationship is that \(a\), \(b\), and \(c\) are in GP.

Revision Table: Understanding Progressions

Additional Information: Links Between Progressions

This problem highlights the connection between HP and GP through AP. When terms are in HP, their reciprocals are in AP. The condition for AP can then be algebraically manipulated to reveal relationships that might correspond to other progressions like GP, as we saw in this case resulting in \(b^2 = ac\).

Understanding the definitions and key properties of AP, GP, and HP is crucial for solving problems involving these sequences. The mean relationships are particularly useful:

• Arithmetic Mean (AM) of \(a\) and \(c\) is \(\frac{a+c}{2}\). If \(a, b, c\) are in AP, then \(b\) is the AM of \(a\) and \(c\).
• Geometric Mean (GM) of \(a\) and \(c\) is \(\sqrt{ac}\) (for positive \(a, c\)). If \(a, b, c\) are in GP, then \(b\) is the GM of \(a\) and \(c\).
• Harmonic Mean (HM) of \(a\) and \(c\) is \(\frac{2ac}{a+c}\). If \(a, b, c\) are in HP, then \(b\) is the HM of \(a\) and \(c\).

The relationship between AM, GM, and HM for two positive numbers \(a\) and \(c\) is \(AM \ge GM \ge HM\).

In this problem, the specific structure of the terms in HP \((a+b, 2b, b+c)\) led to a condition on \(a, b, c\) themselves, revealing they form a GP.