Problem 110 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (C) is correct.

Step 1: Definition and Properties of G.P.

For numbers \( a, b, c \) in G.P. (Geometric Progression), there exists a common ratio \( r \) such that \( b = ar \) and \( c = ar^2 \).

Step 2: Calculating the Arithmetic Means

The arithmetic mean \( x \) between \( a \) and \( b \) is \( x = \frac{a + b}{2} = \frac{a + ar}{2} \). Similarly, the arithmetic mean \( y \) between \( b \) and \( c \) is \( y = \frac{b + c}{2} = \frac{ar + ar^2}{2} \).

Step 3: Simplification of Arithmetic Means

Simplifying \( x \) and \( y \), we get \( x = \frac{a(1 + r)}{2} \) and \( y = \frac{ar(1 + r)}{2} \).

Step 4: Substitution in Potential Answers

Check each option using these expressions for \( x \) and \( y \). Option (C) requires \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \) or \( \frac{2}{a(1+r)} + \frac{2}{ar(1+r)} \). This becomes \( \frac{2}{a(1+r)}(1 + \frac{1}{r}) = \frac{2}{ar} = \frac{2}{b} \).

Step 5: Verification of Option C

The equation \( \frac{2}{ar} = \frac{2}{b} \) simplifies correctly since \( ar = b \). Thus, the statement is true.

Key Concepts

Arithmetic Mean

Arithmetic Mean, often abbreviated as A.M., is the average or central value of a set of numbers. It is calculated by adding the numbers together and dividing by the count of numbers. In the context of the given problem, if you have two numbers, the arithmetic mean is found by summing up these two numbers and dividing by 2. For example, between numbers like \(a\) and \(b\) in G.P., the A.M. would be \(x = \frac{a + b}{2}\). This operation is straightforward but crucial in deriving relationships between terms in different progressions.
Remember, A.M. is a useful tool in comparing values and understanding their central tendency. In problems involving sequences, A.M. helps establish links between different terms and their respective arithmetic relationships.

Common Ratio

The concept of a common ratio is fundamental when dealing with geometric progressions (G.P.). A geometric progression is a sequence where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. This concept is incredibly significant in identifying and solving problems related to G.P.

• For a sequence where \(a, b, c\) are in geometric progression, the common ratio \(r\) can be expressed as the ratio of successive terms. Thus, \(b = ar\) and \(c = ar^2\).

Understanding the common ratio allows us to explore the spatial relationship between elements in the sequence and to develop equations that describe this progression. Recognizing this relationship is vital in not only solving the exercise but also in making quick assessments of whether a set of terms could form a geometric progression.

Mathematical Problem Solving

Mathematical problem solving involves a series of methodical steps to arrive at a conclusion or solution. In this specific problem of determining the validity of different statements, we analyze and simplify expressions using algebraic manipulation and knowledge of sequences.

• Start by understanding the properties of the sequence, such as G.P., to write down expressions for terms using the common ratio.
• Then, calculate the arithmetic means, simplifying where possible to create manageable equations.
• Finally, substitute these simplified expressions into the various options provided to verify which, if any, satisfy the original conditions.

This process underscores key problem-solving strategies such as substitution, elimination, and logical reasoning to deduce the correct result. Problem-solving in mathematics requires comprehension of abstract concepts and their application in various scenarios to forge a path to the solution.