Question

If \(a, b, c\) are in \(\mathrm{GP}\), then \(\log _{a} d, \log _{b} d, \log _{c} d\) are in : (a) \(\mathrm{AP}\) (b) \(\mathrm{HP}\) (c) GP (d) none of these

Short answer

a) Arithmetic Progression (AP) b) Harmonic Progression (HP) c) Geometric Progression (GP) d) None of the above Solution: The correct answer is (b) Harmonic Progression (HP).

Step-by-step solution

Write the given information and recall the definitions.

We are given that \(a\), \(b\), and \(c\) are in a geometric progression (GP), and we need to find the sequence type of \(\log_{a}d\), \(\log_{b}d\), and \(\log_{c}d\). Recall that: - In a geometric progression, if \(a\), \(b\), and \(c\) are consecutive terms, then \(b = ar\) and \(c = br = ar^2\) for some constant \(r\). - In an arithmetic progression (AP), if \(a\), \(b\), and \(c\) are consecutive terms, then the differences between consecutive terms are equal, i.e., \(b - a = c - b\). - In a harmonic progression (HP), if \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\) are consecutive terms, then they are in AP. - In a geometric progression (GP), if \(a\), \(b\), and \(c\) are consecutive terms, then the ratios between consecutive terms are equal, i.e., \(\frac{b}{a} = \frac{c}{b}\).

Use the logarithm properties.

Using the properties of logarithms, we can rewrite the given terms as: \(\log_{a}d = \frac{\log d}{\log a}\), \(\log_{b}d = \frac{\log d}{\log b}\), and \(\log_{c}d = \frac{\log d}{\log c}\). Now, we'll analyze each option to see which one describes the sequence type of these terms.

Analyze option (a) - Arithmetic Progression (AP).

For the terms to be in AP, we'll check if \( \frac{\log d}{\log b} - \frac{\log d}{\log a} = \frac{\log d}{\log c} - \frac{\log d}{\log b}\). Simplifying, we get: \(\frac{\log d}{\log a} - \frac{\log d}{\log b} = \frac{\log d}{\log b} - \frac{\log d}{\log c}\). Divide by \(\log{d}\), \(\frac{1}{\log a} - \frac{1}{\log b} = \frac{1}{\log b} - \frac{1}{\log c}\). But this is the condition for an HP, not for an AP. Thus, option (a) is incorrect.

Analyze option (b) - Harmonic Progression (HP).

We have already found that the terms satisfy the condition for an HP: \(\frac{1}{\log a} - \frac{1}{\log b} = \frac{1}{\log b} - \frac{1}{\log c}\). Therefore, \(\log_{a}d\), \(\log_{b}d\), and \(\log_{c}d\) are in a harmonic progression. Option (b) is the correct answer.

Verify that options (c) and (d) are incorrect.

We have already found that the terms are in an HP, but it's essential to verify if options (c) and (d) are incorrect. To be a GP, we need to check if \(\frac{\frac{\log d}{\log b}}{\frac{\log d}{\log a}} = \frac{\frac{\log d}{\log c}}{\frac{\log d}{\log b}}\). Simplify to get: \(\frac{\log a}{\log b} = \frac{\log b}{\log c}\). But we know that \(a, b, c\) are in GP, so \(\frac{b}{a} = \frac{c}{b} \implies \frac{\log b}{\log a} = \frac{\log c}{\log b}\). This contradicts the condition for a GP. Therefore, option (c) is incorrect. Since we have found a correct answer in option (b), option (d) must also be incorrect. The correct answer is (b) \(\mathrm{HP}\).

Key concepts

Logarithms

Logarithms are a vital concept in mathematics that often simplify complex operations into manageable ones. When studying logarithms, remember that they are essentially inverse operations to exponentiation. For example, if you have \(a^x = b\), a logarithm can express that relationship as \(x = \log_a b\). This function helps scientists and mathematicians scale large numbers or measure exponential growth more cleanly.

Some core properties of logarithms include:

• Product Rule: \(\log_b (mn) = \log_b m + \log_b n\)
• Quotient Rule: \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\)
• Power Rule: \(\log_b (m^n) = n \log_b m\)

This set of rules is fundamental when manipulating logarithmic expressions like \(\log_a d, \log_b d, \log_c d\) to find if they form any specific progression. Applying these properties may reveal patterns across complex expressions, which is essential in identifying sequences like arithmetic, harmonic, or geometric progressions.

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers derived from an arithmetic progression but applied to their reciprocals. Simply put, if the reciprocals of a sequence are in arithmetic progression (AP), then the sequence itself is in harmonic progression (HP).

For instance, if you have numbers \(a\), \(b\), and \(c\), they are in HP if \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in AP. This implication offers an alternate view, providing more angles to analyze and interpret progressions.

• If the differences in the reciprocals are uniform, it suggests the underlying numbers follow an HP. This can be written as:


• \(\frac{1}{a} - \frac{1}{b} = \frac{1}{b} - \frac{1}{c}\)

This formula helps determine that logarithmic terms like \(\log_a d, \log_b d, \log_c d\) form an HP, as illustrated in the exercise solution. Recognizing this pattern can simplify complex problems into understandable segments.

Arithmetic Progression (AP)

Arithmetic progression (AP) is where you find each term by adding/subtracting a constant difference to/from the previous one. The simplicity of AP makes it an ideal starting point for understanding other progressions like GP and HP.

The general format of an arithmetic progression is \(a, a+d, a+2d, \ldots\), with 'd' being the common difference.

• Key Feature: The uniform difference between terms
• Typical Expression: \(b - a = c - b = d\)

In terms of logarithms, to test if numbers like \(\log_a d, \log_b d, \log_c d\) are in AP, we calculate the difference of their values. However, when using logarithms, sometimes these terms align to form an HP, revealing how intricate and interdependent these mathematical concepts can be.

Understanding how these sequences relate can significantly advance the comprehension of mathematical patterns and relationships.