Problem 110 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are in G.P. and \(x\) is the A.M. between \(a\) and \(b, y\) the A.M. between \(b\) and \(c\), then (A) \(\frac{a}{x}+\frac{c}{y}=1\) (B) \(\frac{a}{x}+\frac{c}{y}=2\) (C) \(\frac{1}{x}+\frac{1}{y}=\frac{2}{b}\) (D) None of these

Step-by-Step Solution

Answer

(B) \( \frac{a}{x} + \frac{c}{y} = 2 \).

Step 1: Define Geometric Progression

In a geometric progression (G.P.), the ratio of consecutive terms is constant. If \( a, b, c \) are in G.P., we can express them as \( a, ar, ar^2 \) where \( r \) is the common ratio. Hence, \( b = ar \) and \( c = ar^2 \).

Step 2: Calculate Arithmetic Mean (AM)

The arithmetic mean (AM) of two numbers \( x \) and \( y \) is given by \( \frac{x+y}{2} \). For \( a \) and \( b \), the AM \( x = \frac{a+b}{2} = \frac{a + ar}{2} \). For \( b \) and \( c \), the AM \( y = \frac{b+c}{2} = \frac{ar + ar^2}{2} \).

Step 3: Simplify the Expressions for x and y

Since \( b = ar \), substitute \( b \) in \( x \) and \( y \).For \( x \):\[ x = \frac{a + ar}{2} = \frac{a(1 + r)}{2} \]For \( y \):\[ y = \frac{ar + ar^2}{2} = \frac{ar(1 + r)}{2} \]

Step 4: Evaluate the Given Equation Options

We need to evaluate the options using \( \frac{a}{x} + \frac{c}{y} \).Calculate \( \frac{a}{x} \):\[ \frac{a}{x} = \frac{a}{\frac{a(1+r)}{2}} = \frac{2}{1+r} \]Calculate \( \frac{c}{y} \):\[ \frac{c}{y} = \frac{ar^2}{\frac{ar(1+r)}{2}} = \frac{2r}{1+r} \]Add the two results:\[ \frac{a}{x} + \frac{c}{y} = \frac{2}{1+r} + \frac{2r}{1+r} = \frac{2 + 2r}{1+r} = 2 \]

Step 5: Determine the Correct Answer

From our calculations, \( \frac{a}{x} + \frac{c}{y} = 2 \). Therefore, the correct option is (B).

Key Concepts

Arithmetic Mean

The arithmetic mean (AM) is a fundamental concept in mathematics used to find the central value of a set of numbers. It is often referred to as the average and is calculated by adding all the numbers in a data set and dividing by the count of numbers. In simpler terms, the arithmetic mean helps to find the middle ground in a series of numbers, smoothing out variations to provide a clearer picture.
For example, if you have two numbers, like in the original exercise where it's between terms in a geometric progression: the mean between two numbers \(a\) and \(b\) is \(\frac{a+b}{2}\). This formula means you add the two numbers together and then divide by two to find their average.
In geometric progressions, even though the numbers follow a multiplicative sequence, arithmetic means provide insight into how the sequence distributes around certain points. This understanding is crucial in many mathematical problems and real-world applications, like statistics, and even analyzing financial data.

Problem Solving

Problem-solving is at the heart of mathematics and involves a range of skills, such as understanding the problem, devising a plan, carrying out the plan, and evaluating the solution for accuracy and efficiency.
In the context of sequences and series, solving a problem typically involves recognizing patterns (such as a geometric progression) and applying relevant mathematical concepts (like calculating arithmetic means or common ratios).
Let's consider the original problem again. We break it down into manageable parts, which is a key problem-solving strategy:

• Recognize that \(a, b, c\) form a geometric sequence and express them in terms of the common ratio \(r\).
• Calculate the arithmetic means as required by the problem's conditions.
• Use algebraic methods to manipulate and simplify expressions to taste out each given equation. This process of simplifying and comparing helps us effectively tackle and answer the original query.

Problem-solving involves both logical analysis and creativity, allowing you to apply foundational knowledge to different kinds of mathematical problems.

Sequences and Series

Sequences and series are a central theme in mathematics and play a crucial role in understanding patterns of numbers. A sequence is an ordered list of numbers, and a series is the sum of terms of a sequence. Each sequence follows a specific rule, guiding how each term relates to the preceding one.
In the original exercise, we dealt with a **geometric progression** (a specific type of sequence), where each term after the first is found by multiplying the previous term by a constant known as the "common ratio". If \(a, b, c\) are in a geometric progression, then we can write them as \(a, ar, ar^2\), assuming \(r\) is the common ratio.
Understanding sequences and series is not just about recognizing these patterns but also using mathematical tools to work with them, such as finding the arithmetic mean or identifying relationships between terms.

• In geometric sequences, knowing the first term and the common ratio allows you to determine any term in the sequence.
• Series let you sum parts of a sequence, revealing cumulative sums that can have practical applications, from finance to physics.

By mastering these concepts, you're establishing a strong foundation in mathematics that intersects with various mathematical disciplines and real-world applications.