If three successive terms of as G.P. with commonratio `rgt1` form the sides of a triangle and [r] denotes the integral part of x the `[r]+[-r]=` (A) 0 (B) 1 (C) -1 (D) none of these

Options

A. 0
B. 1
C. -1
D. none of these

Explanation

To solve the problem step by step, we need to analyze the conditions given and derive the necessary values.

### Step 1: Define the terms in the G.P.
Let the three successive terms of the geometric progression (G.P.) be:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)

### Step 2: Apply the triangle inequality
For these terms to form the sides of a triangle, they must satisfy the triangle inequality:
1. \( a + ar > ar^2 \)
2. \( a + ar^2 > ar \)
3. \( ar + ar^2 > a \)

### Step 3: Simplify the inequalities
Let's simplify each of these inequalities:

1. **First Inequality**:
\[ a + ar > ar^2 \implies a(1 + r) > ar^2 \implies 1 + r > r^2 \implies r^2 - r - 1 < 0 \]

2. **Second Inequality**:
\[ a + ar^2 > ar \implies a(1 + r^2) > ar \implies 1 + r^2 > r \implies r^2 - r + 1 > 0 \]
(This inequality is always true for all real \( r \).)

3. **Third Inequality**:
\[ ar + ar^2 > a \implies a(r + r^2) > a \implies r + r^2 > 1 \implies r^2 + r - 1 > 0 \]

### Step 4: Solve the quadratic inequalities
From the first inequality \( r^2 - r - 1 < 0 \):
- The roots of the equation \( r^2 - r - 1 = 0 \) are:
\[ r = \frac{1 \pm \sqrt{5}}{2} \]
- The inequality \( r^2 - r - 1 < 0 \) holds for:
\[ \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]

From the third inequality \( r^2 + r - 1 > 0 \):
- The roots of the equation \( r^2 + r - 1 = 0 \) are:
\[ r = \frac{-1 \pm \sqrt{5}}{2} \]
- The inequality \( r^2 + r - 1 > 0 \) holds for:
\[ r < \frac{-1 - \sqrt{5}}{2} \quad \text{or} \quad r > \frac{-1 + \sqrt{5}}{2} \]

### Step 5: Combine the ranges
Since \( r > 1 \) is given, we focus on the positive range:
- The valid range for \( r \) is:
\[ 1 < r < \frac{1 + \sqrt{5}}{2} \]

### Step 6: Calculate the integral part
Now, we need to find the integral part of \( r \):
- Since \( r > 1 \) and \( r < \frac{1 + \sqrt{5}}{2} \approx 1.618 \), the integral part \( [r] = 1 \).

### Step 7: Calculate \([r] + [-r]\)
- Here, \([-r]\) is the integral part of \(-r\):
- Since \( r \) is between 1 and 1.618, \(-r\) will be between -1.618 and -1.
- Thus, \([-r] = -2\).

Finally, we calculate:
\[ [r] + [-r] = 1 + (-2) = -1 \]

### Final Answer
Thus, the answer is \(-1\).