If a,b,c are 3 distinct terms in G.P. and are also the lengths of sides of a triangle, then number of integral values of the common ratio of that G.P. is:

Options

A. 0
B. 1
C. 2
D. Infinitely many

Explanation

To solve the problem, we need to find the number of integral values of the common ratio \( r \) of a geometric progression (G.P.) where \( a, b, c \) are the distinct terms of the G.P. and also the lengths of the sides of a triangle.

### Step-by-step Solution:

1. **Define the terms in G.P.**:
Let the three distinct terms in the G.P. be \( a, ar, ar^2 \) where \( a \) is the first term and \( r \) is the common ratio.

2. **Triangle Inequality Conditions**:
For \( a, ar, ar^2 \) to be the sides of a triangle, they must satisfy the triangle inequalities:
- \( a + ar > ar^2 \)
- \( a + ar^2 > ar \)
- \( ar + ar^2 > a \)

3. **Simplifying the inequalities**:
- From \( a + ar > ar^2 \):
\[ a(1 + r) > ar^2 \implies 1 + r > r^2 \implies r^2 - r - 1 < 0 \]
- From \( a + ar^2 > ar \):
\[ a(1 + r^2) > ar \implies 1 + r^2 > r \implies r^2 - r + 1 > 0 \quad \text{(always true for real } r\text{)} \]
- From \( ar + ar^2 > a \):
\[ a(r + r^2) > a \implies r + r^2 > 1 \implies r^2 + r - 1 > 0 \]

4. **Analyzing the inequalities**:
- The first inequality \( r^2 - r - 1 < 0 \) can be solved using the quadratic formula:
\[ r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{1 \pm \sqrt{5}}{2} \]
This gives us the roots \( r_1 = \frac{1 - \sqrt{5}}{2} \) and \( r_2 = \frac{1 + \sqrt{5}}{2} \). The inequality \( r^2 - r - 1 < 0 \) holds between these roots:
\[ \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]

5. **Second inequality**:
- The second inequality \( r^2 + r - 1 > 0 \) is always satisfied for real \( r \).

6. **Third inequality**:
- The third inequality \( r^2 + r - 1 > 0 \) can be solved similarly:
\[ r = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]
This gives us the roots \( r_3 = \frac{-1 - \sqrt{5}}{2} \) and \( r_4 = \frac{-1 + \sqrt{5}}{2} \). The inequality holds outside these roots.

7. **Finding the intersection**:
- We need to find the intersection of the intervals:
- From \( r^2 - r - 1 < 0 \): \( \left( \frac{1 - \sqrt{5}}{2}, \frac{1 + \sqrt{5}}{2} \right) \)
- From \( r^2 + r - 1 > 0 \): \( (-\infty, \frac{-1 - \sqrt{5}}{2}) \cup \left( \frac{-1 + \sqrt{5}}{2}, \infty \right) \)
- The intersection of these intervals gives:
\[ \frac{-1 + \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]

8. **Finding integral values**:
- The approximate values of the bounds are:
- \( \frac{-1 + \sqrt{5}}{2} \approx 0.618 \)
- \( \frac{1 + \sqrt{5}}{2} \approx 2.618 \)
- The integral values of \( r \) in this range are \( 1 \) and \( 2 \).

9. **Conclusion**:
- Therefore, the number of integral values of the common ratio \( r \) is **2**.