if `a_(1),a_(2),a_(3)……,a_(12)` are in A.P and
`Delta_(1)= |{:(a_(1)a_(5),,a_(1),,a_(2)),(a_(2)a_(6),,a_(2),,a_(3)),(a_(3)a_(7),,a_(3),,a_(4)):}| Delta_(2)= |{:(a_(2)a_(10),,a_(2),,a_(3)),(a_(3)a_(11),,a_(3),,a_(4)),(a_(4)a_(12),,a_(4),,a_(5)):}|`
then `Delta_(1):Delta_(2)= "_____"`

Explanation

To solve the problem, we need to find the ratio of the determinants \( \Delta_1 \) and \( \Delta_2 \) given that \( a_1, a_2, a_3, \ldots, a_{12} \) are in Arithmetic Progression (A.P.).

### Step 1: Express the terms in A.P.
Since the terms \( a_1, a_2, a_3, \ldots, a_{12} \) are in A.P., we can express them as:
- \( a_1 = a \)
- \( a_2 = a + d \)
- \( a_3 = a + 2d \)
- \( a_4 = a + 3d \)
- \( a_5 = a + 4d \)
- \( a_6 = a + 5d \)
- \( a_7 = a + 6d \)
- \( a_8 = a + 7d \)
- \( a_9 = a + 8d \)
- \( a_{10} = a + 9d \)
- \( a_{11} = a + 10d \)
- \( a_{12} = a + 11d \)

### Step 2: Calculate \( \Delta_1 \)
The determinant \( \Delta_1 \) is given by:
\[ \Delta_1 = \begin{vmatrix} a_1 a_5 & a_1 & a_2 \\ a_2 a_6 & a_2 & a_3 \\ a_3 a_7 & a_3 & a_4 \end{vmatrix} \]
Substituting the values:
\[ \Delta_1 = \begin{vmatrix} a(a + 4d) & a & a + d \\ (a + d)(a + 5d) & a + d & a + 2d \\ (a + 2d)(a + 6d) & a + 2d & a + 3d \end{vmatrix} \]

### Step 3: Simplify \( \Delta_1 \)
Calculating the first column:
- \( a(a + 4d) = a^2 + 4ad \)
- \( (a + d)(a + 5d) = a^2 + 6ad + d^2 \)
- \( (a + 2d)(a + 6d) = a^2 + 8ad + 2d^2 \)

Thus, we have:
\[ \Delta_1 = \begin{vmatrix} a^2 + 4ad & a & a + d \\ a^2 + 6ad + d^2 & a + d & a + 2d \\ a^2 + 8ad + 2d^2 & a + 2d & a + 3d \end{vmatrix} \]

### Step 4: Calculate \( \Delta_2 \)
The determinant \( \Delta_2 \) is given by:
\[ \Delta_2 = \begin{vmatrix} a_2 a_{10} & a_2 & a_3 \\ a_3 a_{11} & a_3 & a_4 \\ a_4 a_{12} & a_4 & a_5 \end{vmatrix} \]
Substituting the values:
\[ \Delta_2 = \begin{vmatrix} (a + d)(a + 9d) & a + d & a + 2d \\ (a + 2d)(a + 10d) & a + 2d & a + 3d \\ (a + 3d)(a + 11d) & a + 3d & a + 4d \end{vmatrix} \]

### Step 5: Simplify \( \Delta_2 \)
Calculating the first column:
- \( (a + d)(a + 9d) = a^2 + 10ad + 9d^2 \)
- \( (a + 2d)(a + 10d) = a^2 + 12ad + 20d^2 \)
- \( (a + 3d)(a + 11d) = a^2 + 14ad + 33d^2 \)

Thus, we have:
\[ \Delta_2 = \begin{vmatrix} a^2 + 10ad + 9d^2 & a + d & a + 2d \\ a^2 + 12ad + 20d^2 & a + 2d & a + 3d \\ a^2 + 14ad + 33d^2 & a + 3d & a + 4d \end{vmatrix} \]

### Step 6: Find the ratio \( \Delta_1 : \Delta_2 \)
After simplifying both determinants, we find that:
\[ \Delta_1 = -2d^4 \quad \text{and} \quad \Delta_2 = -2d^4 \]

Thus, the ratio is:
\[ \frac{\Delta_1}{\Delta_2} = \frac{-2d^4}{-2d^4} = 1 \]

### Final Answer
\[ \Delta_1 : \Delta_2 = 1 : 1 \]