Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(b_{1}, b_{2}\) and \(b_{3}\left(b_{1}>0\right)\) are three successive terms of a G.P. with common ratio \(r\), the value of \(r\) for which the inequality \(b_{3}>4 b_{2}-3 b_{1}\) holds, is given by (A) \(r>3\) (B) \(r<1\) (C) \(r=2.5\) (D) \(r=1.7\)

Step-by-Step Solution

Answer

The correct options are (A) \(r>3\) and (B) \(r<1\).

Step 1: Define the Geometric Progression

Recall that in a geometric progression (G.P.), the terms have a common ratio \( r \). Given the terms \( b_1, b_2, b_3 \), they can be expressed as \( b_2 = b_1 \cdot r \) and \( b_3 = b_2 \cdot r = b_1 \cdot r^2 \).

Step 2: Set Up the Inequality

According to the problem, we have the inequality \( b_3 > 4b_2 - 3b_1 \). Substitute the expressions of \( b_2 \) and \( b_3 \) in terms of \( b_1 \) and \( r \) into the inequality: \( b_1 \cdot r^2 > 4(b_1 \cdot r) - 3b_1 \).

Step 3: Simplify the Inequality

Simplify the inequality by factoring out \( b_1 \) from every term: \( b_1(r^2 - 4r + 3) > 0 \). Since \( b_1 > 0 \), the inequality reduces to \( r^2 - 4r + 3 > 0 \).

Step 4: Solve the Quadratic Inequality

Solve the quadratic equation \( r^2 - 4r + 3 = 0 \) to find its roots. Use the quadratic formula: \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a=1, b=-4, c=3 \). The roots are \( r = 3 \) and \( r = 1 \).

Step 5: Determine Intervals that Satisfy the Inequality

The quadratic expression \( r^2 - 4r + 3 > 0 \) is satisfied outside the roots \( r = 1 \) and \( r = 3 \), i.e., for \( r < 1 \) and \( r > 3 \). Certify this by testing intervals: \((-\infty, 1)\), \((1, 3)\), and \((3, \infty)\).

Step 6: Choose the Correct Option from Given Choices

Compare the solution intervals with the provided choices. The interval \( r > 3 \) matches option (A), and \( r < 1 \) matches option (B).

Key Concepts

Common Ratio

In a geometric progression (G.P.), understanding the **common ratio** is crucial. It is the constant factor by which each term is multiplied to obtain the next term in the sequence. If we take the terms of G.P. as \( b_1, b_2, b_3, \ldots \), then the common ratio \( r \) is defined such that \( b_2 = b_1 \cdot r \) and \( b_3 = b_2 \cdot r = b_1 \cdot r^2 \).

This relationship shows how each term is derived from the previous one by multiplying with \( r \). It also impresses upon us the repetitive nature of multiplication with a fixed ratio, characteristic of geometric progressions. The common ratio can be any real number, including fractions and negative numbers, which affect the sequence's behavior (whether it grows or shrinks, and its direction).

• If \( r > 1 \), the sequence increases.
• If \( 0 < r < 1 \), the sequence decreases.
• If \( r < 0 \), the sequence alternates in sign.

Recognizing and applying the common ratio correctly helps in solving problems involving G.P., like finding unspecified terms or setting up inequalities.

Quadratic Inequality

A **quadratic inequality** involves expressions with a degree of 2, in which we compare them to a number (or zero) using inequality signs (\(<, \leq, >, \geq\)).

In this exercise, the inequality \( r^2 - 4r + 3 > 0 \) arises while solving for the common ratio \( r \) that meets certain criteria in a G.P. To solve it, follow these steps:

• Identify the corresponding quadratic equation by replacing the inequality with equality: \( r^2 - 4r + 3 = 0 \).
• Find the roots of the equation, which are the values where the expression equals zero. These roots can be found using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -4, c = 3 \).
• Determine the intervals where the quadratic expression is positive or negative by testing intervals defined by these roots.

This method reveals the intervals where the initial quadratic inequality holds true. Understanding quadratic inequalities is essential in algebra, as they frequently appear in various forms of mathematical, scientific, and real-world contexts, dictating the conditions for system constraints.

Roots of a Quadratic Equation

The **roots of a quadratic equation** are the solutions to the equation \( ax^2 + bx + c = 0 \). For this exercise, determining these roots is central to solving the quadratic inequality derived from the G.P. term conditions.

To find the roots, we often use the quadratic formula, \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), as it provides a straightforward way to derive solutions for any quadratic equation. For our case:

• The equation \( r^2 - 4r + 3 = 0 \) corresponds to \( a = 1, b = -4, c = 3 \).

Upon calculation, the roots are found to be \( r = 3 \) and \( r = 1 \). These roots split the number line into intervals where the sign of the quadratic expression can change. They serve as boundaries when testing which intervals satisfy a quadratic inequality.

Understanding how to find and use roots is essential in algebra for solving equations and inequalities. The roots determine critical points, indicating where a parabola, the graph of a quadratic function, intersects the \( r \)-axis.