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Magazine Chapter 8: Problem 59
If three successive terms of a G.P. with common ratio \(r(r>1)\) form the sides of a \(\Delta A B C\) and \([r]\) denotes greatest integer function, then \([r]+[-r]=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these
Short Answer
Expert verified
[-1; the answer is (C) -1.]}
Step by step solution
01
Understanding the properties of a G.P.
In a geometric progression (G.P.), each term is the product of the previous term and the common ratio \(r\). If \(a\), \(ar\), and \(ar^2\) are the sides of triangle \(\Delta ABC\), then they must satisfy the triangle inequality conditions.
02
Apply the triangle inequalities
For the G.P. terms to form a triangle, the following inequalities must hold: \(a + ar > ar^2\), \(a + ar^2 > ar\), and \(ar + ar^2 > a\). Simplifying these gives: \(1 + r > r^2\), \(1 + r^2 > r\), and \(r(1 + r) > 1\).
03
Solving the inequalities
Let's solve the inequalities derived from Step 2. Start with \(1 + r > r^2\). Rearrange this as \(r^2 - r - 1 < 0\). The roots of the quadratic equation \(r^2 - r - 1 = 0\) are \(r = \frac{1 \pm \sqrt{5}}{2}\). The inequality holds between the roots, i.e., \(1 < r < \frac{1 + \sqrt{5}}{2}\). Next, simplify \(1 + r^2 > r\) which is always true for \(r > 1\). Finally, \(r(1 + r) > 1\) simplifies to \(r^2 + r > 1\), which holds for all \(r > 1\).
04
Finding the interval for r
Putting together these results, the only constraint is \(1 < r < \frac{1 + \sqrt{5}}{2}\). The interval for \(r\) is between \(1\) and the golden ratio \(\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.618\).
05
Evaluate [r] and [-r]
Given \(1 < r < \frac{1 + \sqrt{5}}{2}\), the greatest integer function, \([r]\), will be \(1\) since \(r\) takes values a little more than \(1\). For \([-r]\), since \(-r < -1\) and not an integer, the greatest integer less than \(-r\) is \(-2\).
06
Calculate [r]+[-r]
Plugging in values from previous steps, \([r] + [-r] = 1 - 2 = -1\). Thus, the expression evaluates to \(-1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Triangle Inequality Theorem
The Triangle Inequality Theorem is a fundamental concept in geometry. It states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is an essential condition for three segments to form a triangle. In mathematical terms, if a triangle has sides of length \(a\), \(b\), and \(c\), then the following must be true:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
- \(a + ar > ar^2\)
- \(a + ar^2 > ar\)
- \(ar + ar^2 > a\)
Greatest Integer Function
The Greatest Integer Function, also known as the floor function, of a number \(x\) is the largest integer less than or equal to \(x\). It is denoted by \([x]\). For instance, \([3.6] = 3\) and \([-2.3] = -3\). This function rounds down any real number to the nearest integer below it.
In our exercise, we are interested in evaluating \([r]\) and \([-r]\), where \(r\) is a number between 1 and the golden ratio \(\varphi\). Since \(1 < r < 1.618\), for values of \(r\) just slightly above 1, the greatest integer function \([r]\) will be 1 as \(r\) has not yet reached 2.
Conversely, for \([-r]\), since \(-r\) falls between -1.618 and -1, the greatest integer less than \(-r\) will be \(-2\). This results in an evaluation of \([r] + [-r] = 1 - 2 = -1\), which becomes our final answer during step calculations.
In our exercise, we are interested in evaluating \([r]\) and \([-r]\), where \(r\) is a number between 1 and the golden ratio \(\varphi\). Since \(1 < r < 1.618\), for values of \(r\) just slightly above 1, the greatest integer function \([r]\) will be 1 as \(r\) has not yet reached 2.
Conversely, for \([-r]\), since \(-r\) falls between -1.618 and -1, the greatest integer less than \(-r\) will be \(-2\). This results in an evaluation of \([r] + [-r] = 1 - 2 = -1\), which becomes our final answer during step calculations.
Quadratic Equation and Roots
A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\). Its solutions, also called roots, are the values of \(x\) that satisfy the equation. The roots can be found using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Understanding the nature of these roots helps in analyzing intervals for different equations. In our exercise, we came across the inequality \(r^2 - r - 1 < 0\).
The roots of the corresponding equation \(r^2 - r - 1 = 0\) are determined using the quadratic formula:\[r = \frac{1 \pm \sqrt{5}}{2}\]These roots are the conjugate segments \(\frac{1 + \sqrt{5}}{2}\) and \(\frac{1 - \sqrt{5}}{2}\). Evaluating further, \(\frac{1 + \sqrt{5}}{2}\) is approximately 1.618, which is recognized as the golden ratio. The inequality \(r^2 - r - 1 < 0\) is satisfied when \(1 < r < 1.618\), dictating that \(r\) lies between these two roots. This interval is crucial for applying this concept in analyzing whether our terms can form a triangle.
The roots of the corresponding equation \(r^2 - r - 1 = 0\) are determined using the quadratic formula:\[r = \frac{1 \pm \sqrt{5}}{2}\]These roots are the conjugate segments \(\frac{1 + \sqrt{5}}{2}\) and \(\frac{1 - \sqrt{5}}{2}\). Evaluating further, \(\frac{1 + \sqrt{5}}{2}\) is approximately 1.618, which is recognized as the golden ratio. The inequality \(r^2 - r - 1 < 0\) is satisfied when \(1 < r < 1.618\), dictating that \(r\) lies between these two roots. This interval is crucial for applying this concept in analyzing whether our terms can form a triangle.
