Problem 46 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \(-12\) (Option B).

Step 1: Define Terms of a Geometric Progression

In a geometric progression (GP), each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \( r \). The first term is denoted as \( a \), so the terms are \( a, ar, ar^2, ar^3, \ldots \).

Step 2: Formulate Equations

Given the condition that the first two terms (\( a \) and \( ar \)) add up to 12, we have the equation:\[ a + ar = 12 \]Also, the third and fourth terms (\( ar^2 \) and \( ar^3 \)) add up to 48:\[ ar^2 + ar^3 = 48 \]

Step 3: Solve for Common Ratio (r)

From the first equation, factor out \( a \):\[ a(1 + r) = 12 \]From the second equation, factor out \( ar^2 \):\[ ar^2(1 + r) = 48 \]Divide the second equation by the first equation:\[ \frac{ar^2(1 + r)}{a(1 + r)} = \frac{48}{12} \]This simplifies to:\[ r^2 = 4 \]Thus, \( r = 2 \) or \( r = -2 \).

Step 4: Determine Alternating Sign Condition

The problem states the sequence alternates in sign. If \( r = 2 \), the sequence does not alternate as each term is positive or negative as per the sign of \( a \). Thus, we choose \( r = -2 \) where the signs alternate at each step of multiplication.

Step 5: Solve for the First Term (a)

Substitute \( r = -2 \) into the equation derived from the sum of the first two terms:\[ a(1 - 2) = 12 \]This simplifies to:\[ -a = 12 \]Thus, \( a = -12 \).

Key Concepts

Common Ratio

In a geometric progression, the common ratio is fundamental. It is the fixed number that each term is multiplied by to get the next term. This can be a positive or negative number. Given the geometric sequence: \( a, ar, ar^2, ar^3, \ldots \), you can clearly see how the common ratio \( r \) operates.
For example, if the first term \( a = 2 \) and the common ratio \( r = 3 \), the progression would continue as \( 2, 6, 18, 54, \ldots \). Here, each term is obtained by multiplying the previous term by 3.

• Positive Common Ratio: Leads to terms increasing or decreasing in magnitude without sign change.
• Negative Common Ratio: Causes terms to alternate in sign, which is particularly important for solving certain problems.

In the problem provided, you must consider the sign effect and whether the sequence alternates when finding the common ratio.

Alternating Signs

Alternating signs in a geometric progression imply that consecutive terms have opposite signs, such as \( +, -, +, - \), or \( -, +, -, + \). This occurs when the common ratio is negative.
This sequence characteristic impacts how terms are calculated and what solutions are possible.
Consider a situation where the first term \( a \) is positive and the common ratio \( r = -2 \):

• The first term stays as positive \( a \).
• The second term becomes negative \(-ar \).
• The third term reverts back to positive \(ar^2 \), due to multiplication by another negative term.
• This repeating property continues.

In the given exercise, alternating signs dictate that the common ratio must be negative, helping to narrow down the potential values of \( r \).

Term Sum Equation

In problems involving geometric progressions, equations based on the sum of terms play a critical role. These equations allow you to solve for unknowns like the first term, common ratio, or any specific term in the sequence.
For example, if the sum of the first two terms is provided, like in the equation \( a + ar = 12 \), you can extract some useful information. Such equations are central in solving the exercise because:

• They relate different terms of the sequence, showing the dependency on the common ratio.
• They form the basis of solving for multiple unknowns with simultaneous equations.

By combining equations for multiple sums, such as \( ar^2 + ar^3 = 48 \), it becomes possible to simplify and solve using techniques like factoring or substitution. This approach is precisely how we solve for the common ratio \( r \) and first term \( a \) in the exercise.

JEE Main Mathematics

The concepts used in this problem are relevant to JEE Main mathematics, an important entrance examination in India featuring both multiple-choice and numerical questions. Understanding geometric progressions aids in handling complex problems in sequences and series, which is vital for the exam.
The format often involves applications of these principles, such as calculating terms, sums, and employing property-based reasoning like alternating signs.

• Geometric Progressions: Appear frequently as foundational concepts not only in direct questions but also as part of more complex algebraic problems.
• Problem Solving: Taking a mathematical problem like this one and breaking it down into steps—using ratios, sequences, and algebra—is a useful tactic.

Having a thorough understanding of geometric progressions, as applied in exercises like these, equips students for successfully tackling similar problems, enhancing overall preparation for the JEE Main.