Problem 8 - Algebra for JEE ADVANCED

Question

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) be in G.P. such that \(3 a_{1}+7 a_{2}+3 a_{3}-4 a_{5}\) \(=0\). Then common ratio of G.P. can be a. 2 b. \(\frac{3}{2}\) c. \(\frac{5}{2}\) d. \(-\frac{1}{2}\)

Step-by-Step Solution

Answer

The common ratio is \(\frac{3}{2}\).

Step 1: Define Terms in G.P.

A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed number called the common ratio (denoted as \(r\)). In this question, \(a_1, a_2, ..., a_n\) are terms of a G.P., so we have \(a_2 = a_1 \cdot r\), \(a_3 = a_1 \cdot r^2\), ..., \(a_5 = a_1 \cdot r^4\).

Step 2: Substitute G.P. Terms into Equation

Substitute the terms from the G.P. into the equation given: \(3a_1 + 7a_2 + 3a_3 - 4a_5 = 0\). This becomes \(3a_1 + 7(a_1 \cdot r) + 3(a_1 \cdot r^2) - 4(a_1 \cdot r^4) = 0\).

Step 3: Factor Out Common Term

Factor out \(a_1\) from the equation: \(a_1(3 + 7r + 3r^2 - 4r^4) = 0\). Since \(a_1\) cannot be zero in a G.P., focus on solving \(3 + 7r + 3r^2 - 4r^4 = 0\).

Step 4: Rearrange the Equation

Rewrite \(3 + 7r + 3r^2 - 4r^4 = 0\) to make it easier to solve. You get \(-4r^4 + 3r^2 + 7r + 3 = 0\).

Step 5: Test Possible Ratios

Substitute the given options into the simplified equation \(-4r^4 + 3r^2 + 7r + 3 = 0\) to find the correct ratio:- For \(r = 2\): \(-4(2)^4 + 3(2)^2 + 7(2) + 3 = 0\) simplifies to \(-64 + 12 + 14 + 3 = -35\), not zero.- For \(r = \frac{3}{2}\): \(-4\left(\frac{3}{2}\right)^4 + 3\left(\frac{3}{2}\right)^2 + 7\left(\frac{3}{2}\right) + 3\) simplifies to zero.- For \(r = \frac{5}{2}\) and \(r = -\frac{1}{2}\), they result in non-zero values.

Step 6: Confirm the Correct Answer

Since only \(r = \frac{3}{2}\) results in the equation being zero, \(\frac{3}{2}\) is the correct common ratio.

Key Concepts

Common Ratio in a Geometric Progression

In any geometric progression (G.P.), the common ratio plays a fundamental role. It is the constant factor you multiply by to get from one term to the next in a sequence. For example, if your first term is 2 and the common ratio is 3, then the next few terms would be 6, 18, and so on. This ratio is denoted by the symbol \( r \). In our original problem, finding the common ratio was key to solving the equation and identifying the correct option.

To find the common ratio, you follow a simple rule: divide any term by the previous one. It should remain constant throughout the sequence. For instance, if your second term is \( a_2 \) and your first term is \( a_1 \), then \( r = \frac{a_2}{a_1} \).

In the exercise, determining \( r \) was essential because it solved a polynomial equation resulting from expressing terms of the G.P. in terms of \( a_1 \) and \( r \). This systematic approach allows you to unravel even the most convoluted equations involving geometric progressions.

Understanding Sequences

A sequence is an ordered list of numbers. In a geometric sequence, each term is derived by multiplying the previous term by a fixed number called the common ratio. Understanding sequences is crucial because it lays the foundation for investigation and solution methods in math problems.

Geometric sequences, in particular, have distinctive properties that allow us to express each term with reference to the first term. For example, in the progression \( a_1, a_2, a_3, \ldots \), you know that:

• \( a_2 = a_1 \cdot r \)
• \( a_3 = a_1 \cdot r^2 \)
• \( a_4 = a_1 \cdot r^3 \)

This pattern makes it easier to solve related equations.

Recognizing the pattern helps simplify expression in terms of \( r \), as demonstrated in the problem's solution. This skill is essential when you have complex equations that can be simplified using sequence properties.

Solving Polynomial Equations

Polynomial equations are mathematical expressions consisting of variables and coefficients. They involve operations such as addition, subtraction, multiplication, and non-negative integer exponents. Solving these equations forms the core of numerous problem-solving scenarios in mathematics.

In the original exercise, after substituting the terms of the geometric progression into the equation, a polynomial expression was derived: \( -4r^4 + 3r^2 + 7r + 3 = 0 \). Solving this relies on identifying values of \( r \) that satisfy the equation, meaning it resolves to zero.

To determine the correct common ratio from given options, you plug each possibility into the polynomial equation. The correct ratio will make the polynomial equal to zero, confirming it as a valid solution for the problem. This method of testing possible solutions is efficient for handling polynomial equations that arise in mathematical contexts.