Question

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

Short answer

Answer: For the three successive terms of a GP to form the sides of a triangle, their common ratio 'r' must satisfy the following inequality: \(\frac{\sqrt{5}-1}{2}<\mathrm{r}<1\).

Step-by-step solution

Identify the three sides of the triangle

Let \(a, ar\), and \(ar^2\) be the three successive terms of the geometric progression (GP) such that they represent the sides of a triangle.

Determine the triangle inequality conditions

In order for \(a, ar,\) and \(ar^2\) to form a triangle, the triangle inequality must hold true. This means that the following conditions must be met: 1. \(a + ar > ar^2\) 2. \(a + ar^2 > ar\) 3. \(ar + ar^2 > a\)

Simplify the inequality expressions

Let's simplify each inequality by performing algebraic operations. 1. \(a + ar > ar^2 \implies a(1 + r) > ar^2 \implies 1 + r > r^2\) 2. \(a + ar^2 > ar \implies a(1 + r^2) > ar \implies 1 + r^2 > r\) 3. \(ar + ar^2 > a \implies ar(1 + r) > a \implies r(1 + r) > 1\)

Identify the Conditions for Common Ratio \(r\)

Now we focus on the third inequality, \(r(1+r)>1\), as it is a quadratic inequality in terms of \(r\). Considering \(r(1+r)>1\), we can rewrite it as \(r^2 + r - 1 > 0\). The roots of the equation \(r^2 + r - 1 = 0\) are given by: \[r = \frac{-1 \pm \sqrt{1^2 - 4(-1)(-1)}}{2(1)}\] \[r = \frac{-1 \pm \sqrt{5-4}}{2}\] \[r = \frac{-1 \pm (\sqrt{5}-1)}{2}\] So, the inequality \(r^2 + r - 1 > 0\) holds true when \(r\) lies outside the interval formed by the roots, which gives us: \(\frac{\sqrt{5}-1}{2}<\mathrm{r}<1\) Hence, if the common ratio 'r' satisfies the inequality \(\frac{\sqrt{5}-1}{2}<\mathrm{r}<1\), the three successive terms of a GP (a, ar, and ar²) will form the sides of a triangle.

Key concepts

Geometric Progression (GP)

A Geometric Progression (GP), also known as a geometric sequence, is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In simpler terms, each number is a certain multiple of the one before it. For example, in a GP with the first term as 2 and a common ratio of 3, the sequence would be 2, 6, 18, 54, and so on. Each consecutive term can be derived by multiplying the common ratio with the preceding term.

GPs are commonly found in various mathematical problems and real-world scenarios, like the growth of bacteria (if each bacterium splits into two, forming a GP with a common ratio of 2), or in the depreciation of a car's value over time (if it depreciates by a consistent percentage, forming a GP where the common ratio is less than 1). Understanding GPs is crucial for solving problems related to exponential growth or decay.

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental principle in geometry which states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This theorem ensures that with three given line segments, a triangle can only be formed if this condition is satisfied for all three combinations of sides.

The importance of the Triangle Inequality Theorem is monumental because it serves as a basic criterion for the existence of a triangle. This theorem is used not just in theoretical problems or proofs but also in practical applications such as in the field of network topology and in determining the possible distances between points in a space.

Common Ratio of GP

The common ratio of a Geometric Progression (GP) is the factor by which consecutive terms are multiplied to obtain the next term in the sequence. It is denoted by 'r' and is a constant for any given GP. The common ratio plays a pivotal role in defining the nature and behavior of a GP. If the common ratio is greater than 1, the sequence grows larger with each term; if the ratio is between 0 and 1, the sequence decreases; if the ratio is negative, the sequence alternates between positive and negative values.

As seen in the exercise, the specific range for the common ratio determined whether three successive terms of a GP could represent the sides of a triangle in accordance with the Triangle Inequality Theorem. The constraints of the common ratio can yield a variety of outcomes, making it a significant element in the analysis of GPs.

Inequalities in Algebra

Inequalities in algebra are mathematical expressions that show the relationship of being less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) between two expressions. Solving inequalities is about finding the values of variables that make the inequality true. Unlike equations, inequalities do not indicate that two expressions are equal, but rather that one is limited in its size or scope in comparison to the other.

Inequalities are powerful tools for representing conditions and constraints in various fields, such as economics for budget limitations, in engineering for tolerances and specifications, and of course in mathematics for understanding ranges of possible solutions, as in the common ratio of a GP for the sides of a triangle.