Problem 100 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a\) be a fixed real number such that \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\)If \(p, q, \mathrm{r}\) are in A.P. then \(x, y, z\) are in (A) A.P. (B) G.P. (C) H. P (D) None of these

Step-by-Step Solution

Answer

x, y, z are in Harmonic Progression (C).

Step 1: Set up the Equation

Given the condition \( \frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z} = k \). Thus, each term equals some constant \(k\). Rearrange each equation separately: \( a - x = k p x \), \( a - y = k q y \), \( a - z = k r z \).

Step 2: Express x, y, z in terms of a, k, p, q, r

Rewrite each equation: \( a = x(1 + kp) \), \( a = y(1 + kq) \), \( a = z(1 + kr) \). So, we have: \( x = \frac{a}{1 + kp} \), \( y = \frac{a}{1 + kq} \), \( z = \frac{a}{1 + kr} \).

Step 3: Determine Relationships among p, q, r

Since \(p, q, \text{ and } r\) are in arithmetic progression, \( q = \frac{p + r}{2} \). Substitute \( q \) in terms of \( p \) and \( r \) to verify any pattern formed by \( x, y, z \).

Step 4: Identify Pattern of x, y, z

Using the expressions from Step 2, \( x = \frac{a}{1 + k p}, y = \frac{a}{1 + k \left( \frac{p + r}{2} \right)}, z = \frac{a}{1 + k r} \). The denominators form an arithmetic sequence.

Step 5: Conclude the Pattern

Since the terms \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) correspond to an arithmetic progression on the denominators, \(x, y, z\) form a Harmonic Progression.

Key Concepts

Arithmetic Progression

An **arithmetic progression (A.P.)** is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the "common difference." Understanding arithmetic progression is key to solving and comprehending sequences where this consistent interval exists. For instance, in the sequence 2, 4, 6, 8, the common difference is 2.

When you have a sequence in A.P., you can express every term using an algebraic formula. If the first term of the sequence is denoted as \( a_1 \) and the common difference as \( d \), then the \( n^{th} \) term \( a_n \) is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

Knowing this formula not only helps in identifying sequences but also aids in finding any term of the sequence easily. This concept is crucial to tackle problems where relationships within sequences need to be examined, especially when interconnecting arithmetic properties with other forms such as harmonic or geometric progressions.

Algebraic Equations

**Algebraic equations** form the foundation for expressing relationships in mathematics using variables and constants. These equations enable us to manipulate mathematical expressions and find unknown values. For example, you might encounter a linear equation like \( ax + b = 0 \), where \( a \) and \( b \) are constants, and \( x \) is the variable to be solved.

In the given problem, algebraic equations are used to express the relationships between \( x \), \( y \), \( z \), and a fixed real number \( a \) in terms of a constant \( k \). The original equations provided can be rewritten to find each variable as:

• \( x = \frac{a}{1 + kp} \)
• \( y = \frac{a}{1 + kq} \)
• \( z = \frac{a}{1 + kr} \)

This transformation helps in identifying the relationship between different variables when other parameters in the equation follow a particular sequence, such as arithmetic progression, as in the exercise. Algebraic manipulation is therefore indispensable for probing into deeper mathematical concepts and relations.

Sequence and Series

**Sequences and series** are vital concepts in mathematics, manifesting themselves in numerous ways of formally arranging numbers. A sequence is essentially a list of numbers in a specific order. When you sum the elements of a sequence, it becomes a series. Understanding these ideas helps to analyze patterns and relationships between numbers.

In the context of the exercise, understanding sequences is crucial. You encounter different types of sequences, such as arithmetic and harmonic progressions. When the denominators of \( x, y, z \) fall into an arithmetic sequence, it turns the sequence of their reciprocals into a harmonic progression. This is a pivotal discovery for establishing the nature of \( x, y, z \) in the problem.

Recognizing how sequences interact with each other, such as how an arithmetic sequence can lead to a harmonic progression, enriches your toolbox for tackling complex problems. It showcases the interconnectedness of various mathematical concepts, encouraging a deeper exploration of sequences and their fascinating behaviors.