Problem 130 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that the three successive terms of a G.P. will form the sides of a triangle if the common ratio \(r\) satisfies the inequality \(\frac{1}{2}(\sqrt{5}-1)

Step-by-Step Solution

Answer

To prove that three successive terms of a G.P. can form the sides of a triangle if the common ratio \(r\) satisfies the inequality \(\frac{1}{2}(\sqrt{5}-1)r^2\), \(1+r^2>r\), and \(r+r^2>1\). Checking these inequalities for the given range of \(r\), we see that all three are true, which confirms that the three terms indeed form the sides of a triangle for \(\frac{1}{2}(\sqrt{5}-1)

Step 1: Apply Triangle Inequality

In order for these three terms to form a triangle, they must satisfy the Triangle Inequality: 1. \(a + ar > ar^2\) 2. \(a + ar^2 > ar\) 3. \(ar + ar^2 > a\)

Step 2: Simplify Inequalities

Now we need to simplify each inequality to relate them to the common ratio \(r\). 1. \(a + ar > ar^2 \Rightarrow 1 + r > r^2\) 2. \(a + ar^2 > ar \Rightarrow 1 + r^2 > r\) 3. \(ar + ar^2 > a \Rightarrow r + r^2 > 1\)

Step 3: Check the given inequality

We are given the following inequality: \(\frac{1}{2}(\sqrt{5}-1)

Step 4: Verify if the given inequality satisfies the Triangle Inequality

Let's check if this inequality satisfies the three inequalities we derived in Step 2: 1. \(1 + r > r^2\) is true for the given inequality since: \((\frac{1}{2}(\sqrt{5}+1)-1)(\frac{1}{2}(\sqrt{5}-1))= \frac{1}{4}\) 2. \(1 + r^2 > r\) is true in the given range as it's a parabola opening to the left and crossing the x-axis at 0. 3. \(r + r^2 > 1\) is also true for the given inequality since: \(\frac{1}{2}(\sqrt{5}+1) - (\frac{1}{2}(\sqrt{5}+1))^2 = \frac{1}{4}\) Since all three Triangle Inequalities are satisfied for the given inequality, it's proven that the three successive terms of a Geometric Progression will form the sides of a triangle if \(\frac{1}{2}(\sqrt{5}-1)

Key Concepts

Triangle Inequality

The Triangle Inequality is a fundamental concept in geometry that determines whether three lengths can form a triangle. It states:

• The sum of any two sides must be greater than the third side.

This means for sides of lengths \(a\), \(b\), and \(c\) to form a triangle, the following must be true:

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

These inequalities ensure that the three sides can "close" and fully form a triangle without any gaps. Without satisfying these conditions, a triangle is not possible. This principle was applied in the exercise when checking whether the terms of the geometric progression could be sides of a triangle.

Common Ratio

In a geometric progression (G.P.), each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If the first term of the G.P. is \(a\) and the common ratio is \(r\), then the sequence progresses as \(a, ar, ar^2, \ldots\).The common ratio plays a crucial role in the properties of the geometric progression. It affects the "shape" and "growth" of the progression:

• If \(|r| < 1\), the terms get closer to zero.
• If \(|r| = 1\), the terms stay constant or switch signs.
• If \(|r| > 1\), the terms grow unbounded or decrease to infinity in absolute value.

Specifically, the exercise focused on finding a range for \(r\) that ensures the terms could be sides of a triangle, thereby linking arithmetic conditions to geometric constructions.

Inequalities in Mathematics

Understanding inequalities in mathematics is about defining relationships between expressions that are not strictly equal. Inequalities are expressed using signs such as:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

These concepts are crucial for analyzing conditions under which certain mathematical properties hold. For example, inequalities were used in the exercise to figure out the appropriate range for the common ratio \(r\), enabling the terms of a geometric progression to satisfy the Triangle Inequality.By evaluating where \(r\) fits within specific boundaries, it becomes possible to determine the relationships and structural characteristics that enable the geometric progression to mirror the sides of a triangle. Applying inequalities in mathematics allows us to explore boundaries, limits, and feasible solutions for various situations.

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Problem 131

Other exercises in this chapter

Problem 128

If \(a, b, c\) are in G.P. and \(x, y\) respectively be arithmetic means between \(a, b\) and \(b, c\), then prove that \(\frac{a}{x}+\frac{c}{y}=2\) and \(\fra

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Problem 129

If \(a, b, c\) be distinct positive and in G.P. and \(\log _{c} a, \log _{b} c, \log _{a} b\) be in A.P. then show that the common difference of this A.P. is \(

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Problem 131

The fifth term of a G.P. is 81 whereas its second term is 24 . Find the series and sum of its first eight terms.

View solution

Problem 132

The sum of first three of a G.P. is to the sum of the first six terms as \(125: 152\). Find the common ratio of the G.P.

View solution

Key Concepts

Triangle Inequality

The Triangle Inequality is a fundamental concept in geometry that determines whether three lengths can form a triangle. It states:

• The sum of any two sides must be greater than the third side.

This means for sides of lengths \(a\), \(b\), and \(c\) to form a triangle, the following must be true:

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

These inequalities ensure that the three sides can "close" and fully form a triangle without any gaps. Without satisfying these conditions, a triangle is not possible. This principle was applied in the exercise when checking whether the terms of the geometric progression could be sides of a triangle.

Common Ratio

In a geometric progression (G.P.), each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. If the first term of the G.P. is \(a\) and the common ratio is \(r\), then the sequence progresses as \(a, ar, ar^2, \ldots\).The common ratio plays a crucial role in the properties of the geometric progression. It affects the "shape" and "growth" of the progression:

• If \(|r| < 1\), the terms get closer to zero.
• If \(|r| = 1\), the terms stay constant or switch signs.
• If \(|r| > 1\), the terms grow unbounded or decrease to infinity in absolute value.

Specifically, the exercise focused on finding a range for \(r\) that ensures the terms could be sides of a triangle, thereby linking arithmetic conditions to geometric constructions.

Inequalities in Mathematics

Understanding inequalities in mathematics is about defining relationships between expressions that are not strictly equal. Inequalities are expressed using signs such as:

• \(<\): less than
• \(>\): greater than
• \(\leq\): less than or equal to
• \(\geq\): greater than or equal to

These concepts are crucial for analyzing conditions under which certain mathematical properties hold. For example, inequalities were used in the exercise to figure out the appropriate range for the common ratio \(r\), enabling the terms of a geometric progression to satisfy the Triangle Inequality.By evaluating where \(r\) fits within specific boundaries, it becomes possible to determine the relationships and structural characteristics that enable the geometric progression to mirror the sides of a triangle. Applying inequalities in mathematics allows us to explore boundaries, limits, and feasible solutions for various situations.