Problem 24 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a G.P. of alternating positive and negative terms, any term is the A.M. of the next two terms. Then the common ratio is(A) \(-1\) (B) \(-3\) (C) \(-2\) (D) \(\frac{-1}{2}\)

Step-by-Step Solution

Answer

The common ratio is \(-2\).

Step 1: Define the Terms of the G.P.

Consider a geometric progression (G.P.) where the first term is denoted by \(a\), and the common ratio is \(r\). Therefore, the terms of the G.P. are \(a, ar, ar^2, ar^3, \ldots\).

Step 2: Establish the Alternating Term Condition

Since the terms of the G.P. alternate between positive and negative, one possibility is that \(r\) is negative. This means that every other term has opposite sign.

Step 3: Understand the Given Condition of A.M.

According to the problem, any term is the arithmetic mean (A.M.) of the next two terms. Therefore, for any term \(ar^n\), the condition \(ar^n = \frac{(ar^{n+1} + ar^{n+2})}{2}\) should be satisfied.

Step 4: Substitute and Simplify the A.M. Condition

Substitute the terms into the A.M. condition: \(ar^n = \frac{ar^{n+1} + ar^{n+2}}{2}\). Multiply through by 2 to clear the fraction: \(2ar^n = ar^{n+1} + ar^{n+2}\). Divide everything by \(ar^n\) to simplify: \(2 = r + r^2\).

Step 5: Solve the Quadratic Equation

Rearrange the equation to form a quadratic equation: \(r^2 + r - 2 = 0\). Factor the quadratic equation: \((r + 2)(r - 1) = 0\). Hence, the solutions for \(r\) are \(r = -2\) or \(r = 1\).

Step 6: Analyze the Solutions

Since the terms alternate in sign, \(r\) cannot be \(1\) because this would not result in alternating signs. Thus, the common ratio \(r\) must be \(-2\).

Key Concepts

Common Ratio

The common ratio is a key characteristic of a geometric progression (G.P.). It is the factor by which each term of the sequence is multiplied to get the subsequent term. In mathematical terms, if the first term of a G.P. is \(a\) and the common ratio is \(r\), then the sequence can be expressed as: \(a, ar, ar^2, ar^3, \ldots\).

Understanding the common ratio is crucial for determining the nature of the sequence. For example, when the common ratio \(r\) is a negative number, it causes the terms to alternate in sign. This property is important in particular problems where certain characteristics of a sequence, such as alternating signs, are specified.

In a G.P. with a condition that involves alternating signs, as seen in the exercise provided, determining the correct common ratio is essential to satisfy all given conditions. Solving the quadratic equation derived from these conditions helps identify potential values for \(r\), and the nature of the sequence guides which solution is correct.

Alternating Sign

An alternating sign in a sequence means that the terms switch signs as they progress. This is typically seen in geometric progressions where the common ratio is a negative number.

In an alternating sequence with positive and negative terms, the sign changes for every subsequent term. For instance, if the first term is positive, the second term will be negative, the third positive, and so on. This phenomenon arises naturally when the sequence's common ratio is negative.

In the problem at hand, we are told the G.P. has alternating positive and negative terms. Therefore, among the possible solutions for the common ratio, only a negative value will result in the required alternation of signs. Considering this, the common ratio must be \(-2\), as it satisfies the condition of alternating signs and fits within the context of the other given problem constraints.

Arithmetic Mean

The arithmetic mean (A.M.) is a fundamental concept often used to find a balance between numbers. It is calculated as the sum of a set of numbers divided by the count of those numbers. The A.M. of two numbers, say \(x\) and \(y\), is \(\frac{x+y}{2}\).

In the context of this exercise, each term of the geometric progression is supposed to be the arithmetic mean of the next two terms. This condition can be expressed mathematically as: for any term \(ar^n\), the equation \(ar^n = \frac{(ar^{n+1} + ar^{n+2})}{2}\) holds true.

By substituting the terms and simplifying, we form an equation that helps reveal the correct common ratio. The requirement for each term to match the A.M. of the next two terms reveals deeper properties of the sequence. In this particular problem, it led us to solve a quadratic equation, which in turn resulted in identifying the appropriate common ratio that maintains the alternating sign pattern.