If `a_(1), a_(2), a_(3)(a_(1)gt 0)` are in G.P. with common ratio r, then the value of r, for which the inequality `9a_(1)+5 a_(3)gt 14 a_(2)` holds, can not lie in the interval

Options

A. `[1, oo)`
B. `[1,9//5]`
C. `[4//5, 1]`
D. `[5//9,1]`

Answer

To solve the problem, we need to analyze the given inequality involving the terms of a geometric progression (G.P.). Let's break it down step by step.

### Step 1: Define the terms in the G.P.
Let the first term be \( a_1 \). Since \( a_1, a_2, a_3 \) are in G.P. with common ratio \( r \):
- \( a_2 = a_1 \cdot r \)
- \( a_3 = a_1 \cdot r^2 \)

### Step 2: Substitute the terms into the inequality
We need to analyze the inequality:
\[ 9a_1 + 5a_3 > 14a_2 \]
Substituting the values of \( a_2 \) and \( a_3 \):
\[ 9a_1 + 5(a_1 r^2) > 14(a_1 r) \]
This simplifies to:
\[ 9a_1 + 5a_1 r^2 > 14a_1 r \]

### Step 3: Factor out \( a_1 \)
Since \( a_1 > 0 \), we can divide both sides by \( a_1 \):
\[ 9 + 5r^2 > 14r \]

### Step 4: Rearrange the inequality
Rearranging gives us:
\[ 5r^2 - 14r + 9 > 0 \]

### Step 5: Find the roots of the quadratic equation
To find the intervals where this inequality holds, we first find the roots of the quadratic equation:
\[ 5r^2 - 14r + 9 = 0 \]
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 5 \), \( b = -14 \), and \( c = 9 \).
- Calculate the discriminant:
\[ D = (-14)^2 - 4 \cdot 5 \cdot 9 = 196 - 180 = 16 \]
- Now, calculate the roots:
\[ r = \frac{14 \pm \sqrt{16}}{2 \cdot 5} = \frac{14 \pm 4}{10} \]
This gives us:
\[ r_1 = \frac{18}{10} = 1.8 \quad \text{and} \quad r_2 = \frac{10}{10} = 1 \]

### Step 6: Analyze the sign of the quadratic
We need to determine where \( 5r^2 - 14r + 9 > 0 \). The roots are \( r = 1 \) and \( r = 1.8 \).

### Step 7: Test intervals
We test the intervals:
1. \( r < 1 \)
2. \( 1 < r < 1.8 \)
3. \( r > 1.8 \)

- For \( r < 1 \) (e.g., \( r = 0 \)):
\[ 5(0)^2 - 14(0) + 9 = 9 > 0 \quad \text{(True)} \]

- For \( 1 < r < 1.8 \) (e.g., \( r = 1.5 \)):
\[ 5(1.5)^2 - 14(1.5) + 9 = 5(2.25) - 21 + 9 = 11.25 - 21 + 9 = -0.75 < 0 \quad \text{(False)} \]

- For \( r > 1.8 \) (e.g., \( r = 2 \)):
\[ 5(2)^2 - 14(2) + 9 = 20 - 28 + 9 = 1 > 0 \quad \text{(True)} \]

### Conclusion
The inequality \( 5r^2 - 14r + 9 > 0 \) holds for:
- \( r < 1 \) and \( r > 1.8 \)

Thus, the value of \( r \) cannot lie in the interval \( (1, 1.8) \).