Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c, d\) are in G.P., then \(\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=\) (A) \((a b+a c+b c)^{2}\) (B) \((a c+c d+a d)^{2}\) (C) \((a b+b c+c d)^{2}\) (D) None of these

Step-by-Step Solution

Answer

(C) \((ab + bc + cd)^2\)

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

A sequence of numbers is said to be in Geometric Progression (G.P.) if the ratio of any two successive terms is always the same. If \(a, b, c, d\) are in G.P., then this implies \(b = ar\), \(c = ar^2\), and \(d = ar^3\) for some common ratio \(r\).

Step 2: Substitute the G.P. terms into the expression

Substitute \(a, b = ar, c = ar^2\) and \(d = ar^3\) into the given expression \((a^2 + b^2 + c^2)(b^2 + c^2 + d^2)\). This becomes:\[(a^2 + (ar)^2 + (ar^2)^2)((ar)^2 + (ar^2)^2 + (ar^3)^2)\].

Step 3: Simplify the expression

Simplify each part inside the brackets:- \(a^2 + a^2r^2 + a^2r^4 = a^2(1 + r^2 + r^4)\)- \(a^2r^2 + a^2r^4 + a^2r^6 = a^2r^2(1 + r^2 + r^4)\).The entire expression becomes:\[a^2(1 + r^2 + r^4) imes a^2r^2(1 + r^2 + r^4)\].

Step 4: Factor and simplify the expression

The expression now is:\(a^4r^2(1 + r^2 + r^4)^2\). This is essentially \((a^2 imes ar(1 + r^2 + r^4))^2 = (ab + bc + cd)^2\)\ as \((1+r^2+r^4) = ab + bc + cd\)\ by substituting the G.P. terms again.

Step 5: Identify the correct option

The simplified expression matches option (C): \((ab + bc + cd)^2\). Thus, the correct answer is (C).

Key Concepts

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra that helps make complex mathematical problems more manageable. In this context, our task is to break down and reduce expressions to their simplest form so that we can easily see relationships and patterns. When dealing with geometric progression (G.P.) and simplifications, the process can involve replacing terms with equivalent expressions using known mathematical properties.

For instance, take the expression \( (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \) where \( a, b, c, d \) are in G.P. Here, simplifying involves substituting each term using the common ratio steps, which transforms the problem into an easier one using powers.

• Break down complex expressions into simpler components so that they become more understandable.
• Use identities or properties, like G.P. ratios, to replace and reduce terms.
• Watch for opportunities to factor out common terms or simplify through algebraic manipulation.

By simplifying expressions effectively, you can solve problems more efficiently and avoid unnecessary complexity during calculations.

Substitution Method

The substitution method involves replacing variables or expressions with their equivalent to simplify calculations. It's commonly used in equations where a variable or expression can be expressed in terms of another variable. In sequences, particularly geometric progressions, substitution helps in handling powers and coefficients that define the relationships within the sequence.

In the given exercise, substitution occurs when replacing the general terms of the sequence \( a, b, c, \) and \( d \) with specific terms involving a base \( a \) and a common ratio \( r \). Given \( b = ar, c = ar^2, \) and \( d = ar^3 \), substituting these into the expression helps to express everything in terms of \( a \) and \( r \):

• This transformation allows us to work with a single variable, rather than multiple unknowns.
• It highlights the underlying structure of the progression, making simplification or factoring easier.

Replacement by substitution turns a multi-variable problem into a more solvable expression, reducing the cognitive load while performing calculations.

Sequences and Series

Sequences and series are essential mathematical concepts that involve ordered lists of numbers, where the position in the list determines its nature or pattern. A sequence refers to a set of numbers arranged in a specific order, while a series is the sum of a sequence. In a geometric progression (G.P.), each term after the first is the product of the previous term and a fixed, non-zero number called the common ratio.

A geometric sequence can be expressed as: \[ a, ar, ar^2, ar^3, \ldots \] where \( a \) is the initial term and \( r \) is the common ratio. Understanding this sequence enables us to deduce relationships between numbers in the progression, especially when simplifying or solving expressions derived from them.

• Recognize each term's dependence on the common ratio, making the sequence predictable.
• Apply knowledge of the sequence to simplify expressions by identifying repeat patterns.

Understanding the fundamentals of sequences and series simplifies complex algebraic tasks, allowing you to solve expressions efficiently and understand their deeper mathematical relationships.