Problem 81 - A Complete Resource Book in Mathematics for JEE Main

Question

The three successive terms of a G.P. will form the sides of a triangle if the common ratio \(r\) satisfies the inequality (A) \(\frac{\sqrt{3}-1}{2}

Step-by-Step Solution

Answer

Option (B): \( \frac{\sqrt{5}-1}{2}

Step 1: Understanding the G.P. Terms

In a geometric progression (G.P.), successive terms are defined by a common ratio \( r \). If \( a \) is the first term, the second term is \( ar \), and the third term is \( ar^2 \). These terms can represent the sides of a triangle if they satisfy the triangle inequalities.

Step 2: Applying Triangle Inequalities

For any three lengths \( a \), \( b \), and \( c \) to form a triangle, they must satisfy the inequalities: \( a + b > c \), \( a + c > b \), and \( b + c > a \). Here, substitute \( a \), \( ar \), and \( ar^2 \) to get: 1. \( a + ar > ar^2 \)2. \( a + ar^2 > ar \)3. \( ar + ar^2 > a \).

Step 3: Simplifying Inequality 1

Rearranging and simplifying the first inequality, \( a + ar > ar^2 \), by dividing through by \( a \) (assuming \( a > 0 \)), we get: \( 1 + r > r^2 \).

Step 4: Simplifying Inequality 2

The second inequality \( a + ar^2 > ar \) simplifies to \( 1 + r^2 > r \) after dividing through by \( a \).

Step 5: Simplifying Inequality 3

The third inequality \( ar + ar^2 > a \) simplifies to \( r + r^2 > 1 \) after dividing through by \( a \).

Step 6: Solving the Inequalities

The inequalities from Steps 3, 4, and 5 are: 1. \( 1 + r > r^2 \)2. \( 1 + r^2 > r \)3. \( r + r^2 > 1 \).These can be rewritten as:\[ r^2 - r - 1 < 0 \] and \[ r^2 + r - 1 > 0 \]\[ r + r^2 > 1 \] is already covered by the first and second inequalities.

Step 7: Finding the Range for \(r\)

The inequality \( r^2 - r - 1 < 0 \) is satisfied for \( \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \), where \( \frac{1 - \sqrt{5}}{2} \) is irrelevant as it is negative.

Step 8: Choosing the Correct Option

The appropriate option is the range \( \frac{\sqrt{5} - 1}{2} < r < \frac{\sqrt{5} + 1}{2} \) as provided by option (B).

Key Concepts

Triangle Inequality

The triangle inequality theorem is a fundamental concept in geometry that determines if three sides can form a triangle. This theorem states that for any triangle, the length of any two sides must be greater than the length of the remaining side.
For three sides with lengths \( a, b, \) and \( c \), the inequalities that must be satisfied are:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

In the context of geometric progression (G.P.), these inequalities ensure that the sequence of terms formed by the first term, the term multiplied by the common ratio, and the term multiplied by the square of the common ratio, can construct a triangle.

Common Ratio

In a geometric progression, the common ratio \( r \) is a constant that determines the relationship between successive terms. The sequence is formed by multiplying each term by \( r \) to obtain the next.
For example, if the first term is \( a \), then the second term is \( ar \), and the third term is \( ar^2 \). Understanding the common ratio is essential because it dictates the growth or decay of the sequence.
Moreover, when considering the sides of a triangle, the values of \( r \) must satisfy specific inequalities to meet the triangle inequality conditions. Thus, \( r \) not only characterizes the progression but also plays a crucial role in determining when these terms can create a triangle.

Inequality Solutions

Solving inequalities is a critical component in determining the range for the common ratio \( r \) that will allow the terms to form a triangle.
Starting with the inequalities derived for a geometric progression, such as:

• \( r^2 - r - 1 < 0 \)
• \( 1 + r^2 > r \)

one can solve these to find permissible values of \( r \). This requires understanding and manipulating algebraic expressions and inequalities. Simplifying the inequalities helps in finding a range of \( r \) where the progression's terms maintain the necessary properties to form a triangle.
For our exercise, solving these inequalities gives the range \( \frac{\sqrt{5} - 1}{2} < r < \frac{\sqrt{5} + 1}{2} \), the correct choice ensuring the sequence represents triangle sides.

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Key Concepts

Triangle Inequality

The triangle inequality theorem is a fundamental concept in geometry that determines if three sides can form a triangle. This theorem states that for any triangle, the length of any two sides must be greater than the length of the remaining side.
For three sides with lengths \( a, b, \) and \( c \), the inequalities that must be satisfied are:

• \( a + b > c \)
• \( a + c > b \)
• \( b + c > a \)

In the context of geometric progression (G.P.), these inequalities ensure that the sequence of terms formed by the first term, the term multiplied by the common ratio, and the term multiplied by the square of the common ratio, can construct a triangle.

Common Ratio

In a geometric progression, the common ratio \( r \) is a constant that determines the relationship between successive terms. The sequence is formed by multiplying each term by \( r \) to obtain the next.
For example, if the first term is \( a \), then the second term is \( ar \), and the third term is \( ar^2 \). Understanding the common ratio is essential because it dictates the growth or decay of the sequence.
Moreover, when considering the sides of a triangle, the values of \( r \) must satisfy specific inequalities to meet the triangle inequality conditions. Thus, \( r \) not only characterizes the progression but also plays a crucial role in determining when these terms can create a triangle.

Inequality Solutions

Solving inequalities is a critical component in determining the range for the common ratio \( r \) that will allow the terms to form a triangle.
Starting with the inequalities derived for a geometric progression, such as:

• \( r^2 - r - 1 < 0 \)
• \( 1 + r^2 > r \)

one can solve these to find permissible values of \( r \). This requires understanding and manipulating algebraic expressions and inequalities. Simplifying the inequalities helps in finding a range of \( r \) where the progression's terms maintain the necessary properties to form a triangle.
For our exercise, solving these inequalities gives the range \( \frac{\sqrt{5} - 1}{2} < r < \frac{\sqrt{5} + 1}{2} \), the correct choice ensuring the sequence represents triangle sides.