The value of `|(t_1,t_2,t_3),(t_2,t_3,t_4),(t_3,t_4,t_5)|+|(T_1,T_2,T_3),(T_2,T_3,T_4),(T_3,T_4,T_5)|` where `t_i` s are in A.P. (a,d) and `T_k` s are in G.P (A.R) is

Options

A. a + A
B. d + R
C. 0
D. none

Answer

To solve the problem, we need to evaluate the expression:

\[ |(t_1,t_2,t_3),(t_2,t_3,t_4),(t_3,t_4,t_5)| + |(T_1,T_2,T_3),(T_2,T_3,T_4),(T_3,T_4,T_5)| \]

where \( t_i \) are in Arithmetic Progression (A.P.) and \( T_k \) are in Geometric Progression (G.P.).

### Step 1: Define the terms in A.P.
Let the first term of the A.P. be \( a \) and the common difference be \( d \). Then we can express the terms as:
- \( t_1 = a \)
- \( t_2 = a + d \)
- \( t_3 = a + 2d \)
- \( t_4 = a + 3d \)
- \( t_5 = a + 4d \)

### Step 2: Set up the determinant for A.P.
The first determinant can be set up as follows:

\[ D_1 = |(t_1, t_2, t_3), (t_2, t_3, t_4), (t_3, t_4, t_5)| \]

Substituting the values of \( t_i \):

\[ D_1 = |(a, a + d, a + 2d), (a + d, a + 2d, a + 3d), (a + 2d, a + 3d, a + 4d)| \]

### Step 3: Simplify the determinant for A.P.
We can perform row operations to simplify the determinant. Let's subtract the first row from the second and the second row from the third:

\[ D_1 = |(a, a + d, a + 2d), (d, d, d), (d, d, d)| \]

### Step 4: Evaluate the determinant for A.P.
Notice that the second and third rows are identical. Therefore, the determinant is zero:

\[ D_1 = 0 \]

### Step 5: Define the terms in G.P.
Let the first term of the G.P. be \( A \) and the common ratio be \( r \). Then we can express the terms as:
- \( T_1 = A \)
- \( T_2 = Ar \)
- \( T_3 = Ar^2 \)
- \( T_4 = Ar^3 \)
- \( T_5 = Ar^4 \)

### Step 6: Set up the determinant for G.P.
The second determinant can be set up as follows:

\[ D_2 = |(T_1, T_2, T_3), (T_2, T_3, T_4), (T_3, T_4, T_5)| \]

Substituting the values of \( T_k \):

\[ D_2 = |(A, Ar, Ar^2), (Ar, Ar^2, Ar^3), (Ar^2, Ar^3, Ar^4)| \]

### Step 7: Simplify the determinant for G.P.
Again, we can perform row operations. Subtract the first row from the second and the second row from the third:

\[ D_2 = |(A, Ar, Ar^2), (0, Ar^2 - Ar, Ar^3 - Ar^2), (0, Ar^3 - Ar^2, Ar^4 - Ar^3)| \]

This simplifies to:

\[ D_2 = |(A, Ar, Ar^2), (0, Ar( r - 1), Ar^2(r - 1))| \]

### Step 8: Evaluate the determinant for G.P.
Notice that the first column has a zero in the second and third rows, which indicates that the determinant is also zero:

\[ D_2 = 0 \]

### Step 9: Combine the results
Now, we can combine the results of both determinants:

\[ D_1 + D_2 = 0 + 0 = 0 \]

### Final Answer
Thus, the final value is:

\[ \boxed{0} \]