Problem 4 - A Complete Resource Book in Mathematics for JEE Main

Question

Number of increasing geometrical progression(s) with first term unity, such that any three consecutive terms, on doubling the middle become an A.P, is (A) 0 (B) (C) 2 (D) infinity

Step-by-Step Solution

Answer

There is 1 increasing geometric progression.

Step 1: Understanding the Problem

We need to find the number of increasing geometric progression(s) such that for any three consecutive terms, doubling the middle term results in these terms forming an arithmetic progression (A.P). The first term of our geometric progression is unity, which means it is 1.

Step 2: Define the Geometric Progression (G.P)

Let the first term be 1, and the common ratio be \( r \). The terms of the geometric progression are 1, \( r \), \( r^2 \), \( r^3 \), etc.

Step 3: Set the A.P Condition for Consecutive Terms

We want the terms 1, \( r \), and \( r^2 \) to satisfy the condition for being in arithmetic progression after doubling the middle term: 1, \( 2r \), and \( r^2 \) should form an A.P.

Step 4: Formulate the A.P Condition

For 1, \( 2r \), and \( r^2 \) to be in A.P, the difference between consecutive terms should be equal. Therefore, the equation is: \( 2r - 1 = r^2 - 2r \).

Step 5: Solve the Equation

Simplify the equation from the previous step: \( 4r = r^2 + 1 \). Rearrange to form a quadratic equation: \( r^2 - 4r + 1 = 0 \).

Step 6: Solve the Quadratic Equation

Use the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1 \), \( b = -4 \), and \( c = 1 \). Calculate the discriminant: \( (-4)^2 - 4 \times 1 \times 1 = 16 - 4 = 12 \). Solve for \( r \): \( r = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} \).

Step 7: Check Increasing Progression Condition

For the progression to be increasing, the common ratio \( r \) must be greater than 1. The solutions for \( r \) are \( 2 + \sqrt{3} > 1 \) and \( 2 - \sqrt{3} < 1 \). Only \( r = 2 + \sqrt{3} \) is greater than 1.

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P) is a sequence of numbers where the difference between any two consecutive terms is the same. This difference is known as the "common difference". For example, consider the sequence 2, 4, 6, 8. Here, each number is obtained by adding 2 to the previous number. This constant difference is what defines the arithmetic progression.

• Each term in the sequence can be expressed as a linear function of its position number.
• If the first term is denoted as \(a\) and the common difference as \(d\), then the \(n\)-th term is given by: \(a_n = a + (n-1)d\).

In context of the problem, doubling the middle term fits into A.P by ensuring that the newly doubled term keeps the sequence's common difference intact. For it to be a progression, the equation from the problem must be satisfied.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable \(x\) with the general form: \(ax^2 + bx + c = 0\). The solutions to a quadratic equation are known as the roots and can be found using different methods such as factoring, completing the square, or the quadratic formula.
For instance, to use the quadratic formula to find the roots, the equation is solved as follows:

• First, identify the coefficients \(a\), \(b\), and \(c\).
• Calculate the discriminant \(D\) using \(D = b^2 - 4ac\).
• Apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In our particular problem, solving the quadratic equation was crucial to finding the potential common ratios \(r\) that satisfy the condition of forming an A.P when the middle term in a G.P is doubled.

Common Ratio

The common ratio in a geometric progression (G.P) is a crucial element as it defines the constant factor by which each term in the sequence is multiplied to get the next term. For example, in a G.P like 3, 6, 12, 24, the common ratio is 2 because each term is twice the one before it.

• In a G.P, the \(n\)-th term is expressed as \(a_n = ar^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio.
• Understanding the concept of the common ratio helps in determining the behavior of the progression - whether it is increasing, decreasing, or constant.

In the provided problem, identifying a common ratio greater than 1 distinguished an increasing G.P from any other type, leading to finding the appropriate solution for \(r\). Only \(r = 2 + \sqrt{3}\) satisfied the condition of an increasing G.P.