If second third and sixth terms of an A.P. are consecutive terms o a G.P. write the common ratio of the G.P.

Options

A. `(1)/(2)`
B. 1
C. 2
D. 3

Answer

To solve the problem, we need to find the common ratio of the geometric progression (G.P.) formed by the second, third, and sixth terms of an arithmetic progression (A.P.).

### Step-by-Step Solution:

1. **Identify the terms of the A.P.**:
- The nth term of an A.P. is given by the formula:
\[ T_n = a + (n-1)d \]
- Therefore, we can express the second, third, and sixth terms as follows:
- Second term \( T_2 = a + (2-1)d = a + d \)
- Third term \( T_3 = a + (3-1)d = a + 2d \)
- Sixth term \( T_6 = a + (6-1)d = a + 5d \)

2. **Set up the relationship for G.P.**:
- We know that the terms \( T_2, T_3, T_6 \) are consecutive terms of a G.P. This means that:
\[ (T_3)^2 = T_2 \cdot T_6 \]
- Substituting the expressions for \( T_2, T_3, T_6 \):
\[ (a + 2d)^2 = (a + d)(a + 5d) \]

3. **Expand both sides**:
- Left side:
\[ (a + 2d)^2 = a^2 + 4ad + 4d^2 \]
- Right side:
\[ (a + d)(a + 5d) = a^2 + 5ad + ad + 5d^2 = a^2 + 6ad + 5d^2 \]

4. **Set the equations equal**:
\[ a^2 + 4ad + 4d^2 = a^2 + 6ad + 5d^2 \]

5. **Simplify the equation**:
- Cancel \( a^2 \) from both sides:
\[ 4ad + 4d^2 = 6ad + 5d^2 \]
- Rearranging gives:
\[ 4d^2 - 5d^2 = 6ad - 4ad \]
\[ -d^2 = 2ad \]

6. **Factor the equation**:
\[ d(d + 2a) = 0 \]
- This gives us two possible solutions:
- \( d = 0 \) (not useful in this context)
- \( d + 2a = 0 \) which implies \( d = -2a \)

7. **Substitute \( d \) back into the terms**:
- Now we substitute \( d = -2a \) into the terms:
- \( T_2 = a + d = a - 2a = -a \)
- \( T_3 = a + 2d = a - 4a = -3a \)
- \( T_6 = a + 5d = a - 10a = -9a \)

8. **Find the common ratio of the G.P.**:
- The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{T_3}{T_2} = \frac{-3a}{-a} = 3 \]
- We can verify by checking the ratio of the third term to the second term:
\[ r = \frac{T_6}{T_3} = \frac{-9a}{-3a} = 3 \]

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