If the 2nd , 5th and 9th terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

Options

A. `(7)/(4)`
B. `(8)/(5)`
C. `(4)/(3)`
D. 1

Answer

To solve the problem, we need to find the common ratio of the geometric progression formed by the 2nd, 5th, and 9th terms of a non-constant arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Define the terms of the A.P.**:
Let the first term of the A.P. be \( a \) and the common difference be \( d \).
- The 2nd term \( a_2 = a + d \)
- The 5th term \( a_5 = a + 4d \)
- The 9th term \( a_9 = a + 8d \)

2. **Set up the condition for G.P.**:
Since \( a_2, a_5, \) and \( a_9 \) are in G.P., we can use the property of G.P. which states that the square of the middle term is equal to the product of the other two terms:
\[ a_5^2 = a_2 \cdot a_9 \]

3. **Substitute the terms**:
Substitute the values of \( a_2, a_5, \) and \( a_9 \):
\[ (a + 4d)^2 = (a + d)(a + 8d) \]

4. **Expand both sides**:
- Left side:
\[ (a + 4d)^2 = a^2 + 8ad + 16d^2 \]
- Right side:
\[ (a + d)(a + 8d) = a^2 + 8ad + ad + 8d^2 = a^2 + 9ad + 8d^2 \]

5. **Set the equations equal**:
Now, equate both sides:
\[ a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2 \]

6. **Simplify the equation**:
Subtract \( a^2 \) from both sides:
\[ 8ad + 16d^2 = 9ad + 8d^2 \]
Rearranging gives:
\[ 16d^2 - 8d^2 = 9ad - 8ad \]
Simplifying further:
\[ 8d^2 - ad = 0 \]

7. **Factor the equation**:
Factor out \( d \):
\[ d(8d - a) = 0 \]

8. **Solve for \( d \)**:
Since the A.P. is non-constant, \( d \neq 0 \). Thus, we have:
\[ 8d - a = 0 \implies a = 8d \]

9. **Find the common ratio of the G.P.**:
The common ratio \( r \) of the G.P. can be found using:
\[ r = \frac{a_5}{a_2} = \frac{a + 4d}{a + d} \]
Substitute \( a = 8d \):
\[ r = \frac{8d + 4d}{8d + d} = \frac{12d}{9d} = \frac{12}{9} = \frac{4}{3} \]

### Final Answer:
The common ratio of the G.P. is \( \frac{4}{3} \).