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 : (1) `8/5` (2) `4/3` (3) 1 (4) `7/4`

Options

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

Explanation

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

Let’s denote the first term of the A.P. as \( a \) and the common difference as \( d \).

The terms of the A.P. can be expressed as follows:
- The 2nd term: \( a + d \)
- The 5th term: \( a + 4d \)
- The 9th term: \( a + 8d \)

Since these three terms are in G.P., we can use the property of G.P. which states that if \( b \), \( c \), and \( d \) are in G.P., then \( c^2 = b \cdot d \).

Applying this to our terms:
\[ (a + 4d)^2 = (a + d)(a + 8d) \]

Now, let's expand both sides of the equation.

1. **Expand the left side:**
\[ (a + 4d)^2 = a^2 + 8ad + 16d^2 \]

2. **Expand the right side:**
\[ (a + d)(a + 8d) = a^2 + 8ad + ad + 8d^2 = a^2 + 9ad + 8d^2 \]

Now we have the equation:
\[ a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2 \]

3. **Subtract \( a^2 \) from both sides:**
\[ 8ad + 16d^2 = 9ad + 8d^2 \]

4. **Rearranging gives:**
\[ 16d^2 - 8d^2 = 9ad - 8ad \]
\[ 8d^2 = ad \]

5. **Dividing both sides by \( d \) (since \( d \neq 0 \) for a non-constant A.P.):**
\[ 8d = a \]

Now we have \( a = 8d \).

Next, we can find the common ratio \( r \) of the G.P. formed by the terms \( a + d \), \( a + 4d \), and \( a + 8d \).

6. **Substituting \( a \) into the terms:**
- The 2nd term: \( a + d = 8d + d = 9d \)
- The 5th term: \( a + 4d = 8d + 4d = 12d \)
- The 9th term: \( a + 8d = 8d + 8d = 16d \)

7. **Finding the common ratio \( r \):**
\[ r = \frac{\text{5th term}}{\text{2nd term}} = \frac{12d}{9d} = \frac{12}{9} = \frac{4}{3} \]

Thus, the common ratio of the G.P. is \( \frac{4}{3} \).

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