Problem 95 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are positive numbers in G.P. and log \(\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)\) and \(\log \left(\frac{a}{3 b}\right)\) are in A.P. then \(a, b, c\)(A) form the sides of an equilateral triangle (B) form the sides of an isosceles triangle (C) form the sides of a right angled triangle (D) can not form the sides of a triangle

Step-by-Step Solution

Answer

The sides form an equilateral triangle.

Step 1: Understanding Geometric Progression (G.P.)

Since \(a, b, c\) are in geometric progression, this can be expressed as \(b = ar\) and \(c = ar^2\) for some constant \(r\). Hence, \(a, b, c\) are terms of G.P.

Step 2: Setting up Arithmetic Progression (A.P.) condition

For the terms \(\log\left(\frac{5c}{a}\right), \log\left(\frac{3b}{5c}\right), \log\left(\frac{a}{3b}\right)\) to be in A.P., the difference between consecutive terms is constant. Therefore, \(\log\left(\frac{3b}{5c}\right) - \log\left(\frac{5c}{a}\right) = \log\left(\frac{a}{3b}\right) - \log\left(\frac{3b}{5c}\right)\).

Step 3: Simplifying Using Logarithmic Properties

Using properties of logarithms, this equation can be simplified to: \[ \log\left(\frac{3b}{5c} \cdot \frac{a}{5c}\right) = \log\left(\frac{a}{3b} \cdot \frac{5c}{3b}\right) \] which simplifies to: \[ \log\left(\frac{3b \cdot a}{25c^2}\right) = \log\left(\frac{5ac}{9b^2}\right) \] Relaxing \(\log\), the argument on both sides must be the same, i.e., \(\frac{3ab}{25c^2} = \frac{5ac}{9b^2}\).

Step 4: Solving the Equation for \(r\)

Substituting \(b=ar\) and \(c=ar^2\) into \(\frac{3ab}{25c^2} = \frac{5ac}{9b^2}\), transform this to \(\frac{3a(ar)}{25(ar^2)^2} = \frac{5aar^2}{9(ar)^2}\). After simplification, this should lead to a symmetry or common expression for a particular value of \(r\).

Step 5: Checking Triangle Formation Conditions

Looking at the equation found and knowing the conditions for triangles: notice if a solution, such as symmetry or specific relations, implies any of the side lengths can form a specific type of triangle. For example, check if two sides are the same or if a side squared equals the sum of the squares of the other two for a right triangle.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This characteristic difference is called the "common difference." For example, in the sequence 2, 5, 8, 11, the common difference is 3.

When dealing with logarithmic expressions in an A.P., we apply this concept by ensuring the difference between their values remains constant. For instance, if \(\log(a), \log(b), \log(c)\)form an A.P., then the equation \(\log(b) - \log(a) = \log(c) - \log(b)\) holds true. This relationship can be further simplified using logarithmic properties.

• Identifying common differences is key in solving problems involving A.P.
• Simplifying logarithmic expressions in A.P. often involves utilizing properties of logarithms effectively.

Logarithmic Properties

Logarithmic properties, such as the product, quotient, and power rules, are vital in maneuvering through complex equations. Simplifying expressions using these properties can transform a problem into a manageable form.

For any logarithmic equations, you may often use:

• Product Rule: \(\log(ab) = \log(a) + \log(b)\)
• Quotient Rule: \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
• Power Rule: \(\log(a^b) = b\cdot \log(a)\)

In the context of the sequence of logs in the problem, these properties allow converting differences into simpler expressions, aiding in combing the sequence behavior with progression conditions.

Mastery of these logarithmic rules enhances solving efficiency, especially when sequences dovetail into progression themes in mathematical problems.

Triangle Inequality

The triangle inequality theorem is a crucial concept in geometry. It states that in any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the third side.

Symbolically, for a triangle with sides \(a\), \(b\), and \(c\):

• \(a + b > c\)
• \(b + c > a\)
• \(a + c > b\)

When dealing with problems where lengths are in progression, as seen in this exercise, the triangle inequality can help determine the nature of the triangle that sections of a progression may form, such as an equilateral, isosceles, or right triangle.

Ensuring that the inequality holds tight enhances problem-solving by confirming the possibility and type of triangle formation in such geometric contexts.

Geometric Sequences

Geometric sequences (or progressions) are sequences where each term after the first is found by multiplying the previous term by a constant known as the "common ratio." A simple geometric sequence could be 3, 6, 12, 24, where each term is multiplied by the common ratio of 2.

In our exercise, since \(a, b, c\) form a geometric progression, the relationships can be expressed with a common ratio \(r\) as follows:

• \(b = ar\)
• \(c = ar^2\)

By substituting these relationships into conditions or equations provided, you can unravel the precise dynamics between these terms.

Using properties of geometric sequences allows you to analyze growth patterns effectively, making it easier to link sequence behavior with broader mathematical principles, such as those required for solving the given exercise efficiently.