Problem 85 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The first two terms of a geometric progression add up to 12\. the sum of the third and the fourth terms is 48 . If the terms of the geometric progression are alternately positive and negative, then the first term is (a) \(-4\) (b) \(-12\) (c) 12 (d) 4

Step-by-Step Solution

Answer

The first term is \(-12\).

Step 1: Define Geometric Progression Terms

The first term of the geometric progression (GP) is denoted by \( a \), and the common ratio by \( r \). The first term is \( a \), the second term is \( ar \), the third term is \( ar^2 \), and the fourth term is \( ar^3 \).

Step 2: Establish Equations from Given Conditions

We know from the problem statement: \( a + ar = 12 \) and \( ar^2 + ar^3 = 48 \). This forms our system of equations for solving the problem.

Step 3: Simplify Initial Equations

From the first equation, factor out \( a \): \( a(1 + r) = 12 \). From the second equation, factor out \( ar^2 \): \( ar^2(1 + r) = 48 \).

Step 4: Express Common Ratio in Terms of First Term

Divide the second equation by the first equation to eliminate \( a \): \[ \frac{ar^2(1 + r)}{a(1 + r)} = \frac{48}{12} \Rightarrow r^2 = 4 \]Thus, \( r = 2 \) or \( r = -2 \) because terms are alternately positive and negative.

Step 5: Solve for the First Term when \( r = -2 \)

Using \( r = -2 \), substitute in the first equation: \[ a(1 - 2) = 12 \Rightarrow -a = 12 \Rightarrow a = -12 \]Check the condition for \( ar^2 + ar^3 = 48 \) using \( a = -12 \) and \( r = -2 \):\((-12)(-2)^2 + (-12)(-2)^3 = 48.\)This simplifies to:\(48 + 96 = 48,\)which holds true for this scenario.

Key Concepts

Alternating Signs in Geometric Progression

In a geometric progression, the terms can follow a pattern of alternating signs. This means sequential terms alternate between positive and negative values. When dealing with alternating signs:

• The common ratio can be either positive or negative, influencing the sign change between adjacent terms.
• In cases where terms alternate, if one term is positive, the next one is negative, and then back to positive, and so on.

Knowing the terms alternately change signs is crucial to solving the problem, as it directly affects the choice of the common ratio. In this exercise, choosing a negative common ratio established an alternating sign pattern.

Understanding the Common Ratio

The common ratio in a geometric progression (GP) is the factor by which we multiply a term to get the next term. For example, if the common ratio is denoted by \(r\):

• The second term is given by \(ar\), where \(a\) is the first term.
• The third term is \(ar^2\).
• The fourth term is \(ar^3\).

In instances of alternating signs, the common ratio is crucial. Here, we found that the common ratio \(r\) could be ±2. By analyzing both possibilities, we discovered that \(r = -2\) fits the criteria of the progression having alternating signs. Thus, recognizing the negative value here is essential to maintaining the pattern.

Formulating a System of Equations

When facing a problem involving sequences or series, constructing a system of equations is an effective method for finding unknown values. The system of equations arises from the conditions given in the problem statement. For example:

• The sum of the first two terms results in one equation: \(a + ar = 12\).
• The sum of the third and fourth terms results in another: \(ar^2 + ar^3 = 48\).

By solving these equations simultaneously, you can deduce important variables such as \(a\) and \(r\). In our specific scenario, we derived the value of the common ratio \(r\) and then used it to find \(a\). It’s essential to learn how to manipulate and solve such systems of equations, as they are fundamental in linking provided conditions to unknown variables in an organized manner.