Problem 125 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}\) and \(a, b, c\) be in G.P. then prove that \(x, y, z\) are in A.P.

Step-by-Step Solution

Answer

The short answer is: By manipulating the given equation according to the definitions of G.P. and A.P., we are able to write two equations, both equal to \(\frac{\ln{r}}{\ln{a}}\), representing the differences between consecutive terms \(x\), \(y\), and \(z\). This confirms that \(x\), \(y\), and \(z\) are in Arithmetic Progression (A.P.).

Step 1: Rewrite the given equation using logarithm properties

Since \(a^{\frac{1}{x}}=b^{\frac{1}{y}}=c^{\frac{1}{z}}\), we can take the natural logarithm (or any logarithm) on both sides of the equation: \[\ln\left(a^{\frac{1}{x}}\right)=\ln\left(b^{\frac{1}{y}}\right)=\ln\left(c^{\frac{1}{z}}\right)\] Now, using the logarithm property \(n\cdot\log{a}=\log(a^n)\), the equation becomes: \[\frac{1}{x}\ln{a}=\frac{1}{y}\ln{b}=\frac{1}{z}\ln{c}\]

Step 2: Write the condition for G.P.

Since \(a, b, c\) are in G.P., we can write \(b = a\cdot r\) and \(c = b\cdot r = a\cdot r^2\), where \(r\) is the common ratio.

Step 3: Replace \(b\), \(c\) in the equation from Step 1

Substitute \(b = a\cdot r\) and \(c = a\cdot r^2\) in the equation from Step 1: \[\frac{1}{x}\ln{a}=\frac{1}{y}\ln{(a\cdot r)}=\frac{1}{z}\ln{(a\cdot r^2)}\]

Step 4: Simplify the equation

Using the logarithm property \(\log{A\cdot B} = \log{A} + \log{B}\), we can simplify the equation as: \[\frac{1}{x}\ln{a}=\frac{1}{y}(\ln{a}+\ln{r})=\frac{1}{z}(2\ln{a}+2\ln{r})\]

Step 5: Solve for the common difference of \(x, y, z\)

From the equation in Step 4, we can write the following system of equations: \[\frac{1}{y}-\frac{1}{x}=\frac{\ln{r}}{\ln{a}}\] and \[\frac{1}{z}-\frac{1}{y}=\frac{2\ln{r}}{2\ln{a}}=\frac{\ln{r}}{\ln{a}}\] Since these two equations are equal, the difference between consecutive terms is constant, and we conclude that \(x, y, z\) are in Arithmetic Progression (A.P.).

Key Concepts

Geometric Progression

A geometric progression (often abbreviated as G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, if you start with 2 and use a common ratio of 3, the sequence is 2, 6, 18, 54, and so forth.

• Geometric progression is characterized by the property that the ratio of any term to the previous one is constant.
• The general form of a geometric progression can be written as: \[ a, a \cdot r, a \cdot r^2, a \cdot r^3, \ldots \]
• The common ratio \( r \) is found by dividing any term by the previous term.

Understanding how to form and use geometric progressions can help solve many types of mathematical problems, such as those involving exponential growth or decay.
In the context of the given exercise, understanding G.P. allowed us to express terms \(b\) and \(c\) in terms of \(a\) and the common ratio \(r\), which was crucial for proving that the exponents \(x, y, z\) form an arithmetic progression.

Logarithmic Properties

Logarithms are very useful when dealing with exponential equations, particularly when you need to simplify or solve them. In the context of the given problem, several logarithm properties are crucial. First, consider the property that allows you to break down a logarithm of a power:

• \( n \cdot \log{a} = \log(a^n) \) - This is useful when you have exponential terms.
• Another useful property is \( \log(A \cdot B) = \log{A} + \log{B} \), which lets you separate products into sums when working with logs.

In the solution, these properties were key in transforming the initial equation into a more manageable form. By applying them, we can see that both sides of the equation, \(a^{1/x}, b^{1/y}, c^{1/z}\), have their logarithmic expressions simplified and standardized, allowing for direct comparison. This paves the way to solve for relationships (such as arithmetic progressions) within the exponents \(x, y,\) and \(z\).
Logarithmic properties turn complex multiplicative relations into simpler additive ones, instrumental in proving the end result of the exercise.

Common Ratio

The common ratio is a fundamental component of a geometric progression. It is the fixed number by which each term in the sequence is multiplied to get the next term. In equations involving progressions, identifying and working with the common ratio simplifies the problem.

• For a sequence \(a, b, c\) in G.P., the term \(b\) is given by \(b = a \cdot r\), and \(c = b \cdot r = a \cdot r^2\).
• The common ratio \(r\) is crucial to express terms related to each other, allowing us to replace variables in the equations or set equal the exponents.

In our exercise, understanding the common ratio's application was essential to substitute \(b\) and \(c\) correctly, transforming the logarithmic equation into a form that facilitated solving for the arithmetic nature of exponents \(x, y, z\). Working with a common ratio is a powerful tool not only in pure mathematics but in real-world problems, where ratios represent consistent proportional change over a set of data or conditions.