For the A.P given by `a_(1),a_(2). . .a_(n)` . . . ., the equation satisfied is/are

Options

A. `a-(1)+2a_(2)+a_(3)=0`
B. `a_(1)+2a_(2)+a_(3)=0`
C. `a_(1)+3a_(2)-3a_(3)-a_(4)=0`
D. `a_(1)-4a_(2)+6a_(3)-4a_(4)+a_(5)=0`

Answer

To solve the problem, we need to analyze the equations given for the arithmetic progression (A.P) defined by the terms \( a_1, a_2, \ldots, a_n \) where \( a_n = a_1 + (n-1)d \) and \( d \) is the common difference. We will check each option to see if it holds true.

### Step-by-Step Solution:

1. **Understanding the A.P Terms**:
The terms of the A.P can be expressed as:
- \( a_1 = a_1 \)
- \( a_2 = a_1 + d \)
- \( a_3 = a_1 + 2d \)
- \( a_4 = a_1 + 3d \)
- \( a_n = a_1 + (n-1)d \)

2. **Analyzing Option A**:
The equation given is:
\[ a_1 - a_2 + 2a_3 = 0 \]
Substituting the values:
\[ a_1 - (a_1 + d) + 2(a_1 + 2d) = 0 \]
Simplifying:
\[ a_1 - a_1 - d + 2a_1 + 4d = 0 \implies 2a_1 + 3d = 0 \]
This cannot be true for all A.Ps, hence option A is incorrect.

3. **Analyzing Option B**:
The equation given is:
\[ a_1 + a_2 + a_3 = 0 \]
Substituting the values:
\[ a_1 + (a_1 + d) + (a_1 + 2d) = 0 \]
Simplifying:
\[ 3a_1 + 3d = 0 \implies a_1 + d = 0 \]
This cannot hold for all A.Ps, hence option B is incorrect.

4. **Analyzing Option C**:
The equation given is:
\[ a_1 + 3a_2 - 3a_3 - a_4 = 0 \]
Substituting the values:
\[ a_1 + 3(a_1 + d) - 3(a_1 + 2d) - (a_1 + 3d) = 0 \]
Simplifying:
\[ a_1 + 3a_1 + 3d - 3a_1 - 6d - a_1 - 3d = 0 \]
This simplifies to:
\[ 0 = 0 \]
This holds true, hence option C is correct.

5. **Analyzing Option D**:
The equation given is:
\[ a_1 - 4a_2 + 6a_3 - 4a_4 + a_5 = 0 \]
Substituting the values:
\[ a_1 - 4(a_1 + d) + 6(a_1 + 2d) - 4(a_1 + 3d) + (a_1 + 4d) = 0 \]
Simplifying:
\[ a_1 - 4a_1 - 4d + 6a_1 + 12d - 4a_1 - 12d + a_1 + 4d = 0 \]
This simplifies to:
\[ 0 = 0 \]
This holds true, hence option D is also correct.

### Conclusion:
The equations that hold true for the A.P are:
- Option C: \( a_1 + 3a_2 - 3a_3 - a_4 = 0 \)
- Option D: \( a_1 - 4a_2 + 6a_3 - 4a_4 + a_5 = 0 \)