Question

If \(a^{x}, b^{x}\) and \(c^{x}\) are in G.P., then which of the following is/are true? (a) \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in G.P. (b) loga, logb, logc are in G.P. (c) loga, logb, logc are in A.P. (d) \(a, b, c\) are in A.P. (1) a and b (2) a and \(c\) (3) \(\mathrm{b}\) and \(\mathrm{d}\) (4) only a

Short answer

(a) \(a, b, c\) are in geometric progression. (b) \(log(a), log(b), log(c)\) are in geometric progression. (c) \(log(a), log(b), log(c)\) are in arithmetic progression. (d) \(a, b, c\) are in arithmetic progression. Answer: (c) only, \(log(a), log(b), log(c)\) are in arithmetic progression.

Step-by-step solution

Write the general form of a GP sequence

Since \(a^x, b^x\) and \(c^x\) are in G.P., we can write \(b^x = k \times a^x\) and \(c^x = k \times b^x\) for some constant \(k\).

Rewrite the GP equations in terms of logarithms

To analyze \(a, b, c\), we can rewrite our GP relation as: \(log(b^x) = log(k \times a^x)\) and \(log(c^x) = log(k \times b^x)\).

Simplify logarithms using properties of logarithms

Using the logarithm properties \(log(a^b)= b \times log(a)\) and \(log(a \times b) = log(a) + log(b)\), we get: \(x \times log(b) = x \times log(a) + log(k)\) and \(x \times log(c) = x \times log(b) + log(k)\).

Analyze if \(a, b, c\) are in GP, AP or not

Let's now evaluate the given options: (a) If \(a, b, c\) are in G.P., then \(b = n \times a\) and \(c = n^2 \times a\) for some constant \(n\). This doesn't necessarily follow from the above equations. (b) If \(log(a), log(b), log(c)\) are in G.P., then \((log(b))^2 = log(a) \times log(c)\). This doesn't necessarily follow from the above equations. (c) If \(log(a), log(b), log(c)\) are in arithmetic progression (A.P.), then \(2log(b) = log(a) + log(c)\). By dividing the above-logarithmic equations, we get \(\frac{log(c)}{log(b)} = \frac{log(b)}{log(a)}\). By cross-multiplying, we get \((log(b))^2 = log(a) \times log(c)\). This is true. (d) If \(a, b, c\) are in A.P., then \(2b = a+c\). This doesn't necessarily follow from the above equations. From our analysis, only option (c) is correct, so the final answer is (4) only a.

Key concepts

Logarithmic Properties

Logarithmic properties are vital tools that help in simplifying and solving equations involving exponentials. Let's delve into two key properties used in this exercise:

• Power Property: This involves expressing the logarithm of a power. The formula is \( \log(a^b) = b \times \log(a) \). It means that you can move the exponent in front as a multiplier.
• Product Property: This property states that the logarithm of a product is equal to the sum of the logarithms of its factors:\( \log(a \times b) = \log(a) + \log(b) \).

When working with sequences like geometric progressions (G.P.) in this exercise, these properties are used to transform multiplicative relationships (like those in G.P.) into additive ones, making it easier to find connections/analyze sequences. The exercise illustrated how we can use these properties to transition between exponential forms and logarithmic relations effectively.

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference of any two consecutive terms is always the same.

• Definition: In an A.P., the sequence can be expressed as \( a, a+d, a+2d, \ldots \) where \( d \) is the common difference.
• Equation: If a sequence \( a, b, c \) is in A.P., it follows \( 2b = a+c \). This rearranges the relationships between the terms for easy verification of the arithmetic nature.

While analyzing whether a sequence is in A.P. can sometimes be straightforward, it's crucial to carefully evaluate the conditions like the ones in this exercise to ensure the correct understanding of the type of progression at hand. The exercise demonstrated that the sequences of logarithms could also form arithmetic progressions by using logarithmic properties.

Sequences and Series

Sequences and Series form the backbone of understanding various kinds of progression in mathematics. They represent ordered lists of numbers following specific rules.

• Sequence: It is a list of numbers written in a specific order where each number is called a term. For example, in mathematical sequences like arithmetic and geometric, each term follows a defined rule or pattern.
• Series: This is the sum of the terms of a sequence, represented with terms often added together.

Specifically, this exercise dealt with a geometric progression (G.P.) where terms follow a constant ratio, like \( a, ar, ar^2, \ldots \). The challenge was to identify the nature of sequences when transformed through logarithms into another form of sequences. Handling sequences and transitioning between different forms, like from G.P. to A.P., requires using properties like logarithmic equations to discover new relationships. It's crucial to analyze these transformations and relationships accurately, as demonstrated in the exercise.