Let `B_1,C_1 and D_1` are points on `AB,AC and AD` of the parallelogram `ABCD,` such that `vec(AB_1)=k_1vec(AC,) vec(AC_1)=k_2vec(AC) and vec(AD_1)=k_2 vec(AD,)` where `k_1,k_2 and k_3` are scalar.

Options

A. `lambda_(1), lambda_(3) and lambda_(2) ` are in AP
B. `lambda_(1), lambda_(3) and lambda_(2)` are in GP
C. `lambda_(1),lambda_(3) and lambda_(2)` are in HP
D. `lambda_(1)+lambda_(2)+lambda_(3)=0`

Answer

To solve the problem, we need to analyze the given points \( B_1, C_1, \) and \( D_1 \) in the context of the parallelogram \( ABCD \) and the relationships defined by the scalars \( k_1, k_2, \) and \( k_3 \).

### Step-by-Step Solution:

1. **Define the Points**:
Let \( A \) be the origin (0 vector), \( B \), \( C \), and \( D \) be represented by vectors \( \vec{b}, \vec{c}, \) and \( \vec{d} \) respectively. The points \( B_1, C_1, \) and \( D_1 \) are defined as:
\[ \vec{AB_1} = k_1 \vec{AB}, \quad \vec{AC_1} = k_2 \vec{AC}, \quad \vec{AD_1} = k_3 \vec{AD} \]

2. **Express the Vectors**:
Since \( \vec{AB} = \vec{b}, \vec{AC} = \vec{c}, \vec{AD} = \vec{d} \), we can express:
\[ \vec{B_1} = k_1 \vec{b}, \quad \vec{C_1} = k_2 \vec{c}, \quad \vec{D_1} = k_3 \vec{d} \]

3. **Using the Parallelogram Properties**:
In a parallelogram, we have:
\[ \vec{C} = \vec{A} + \vec{B} + \vec{D} \implies \vec{C} = \vec{b} + \vec{d} \]

4. **Apply Triangle Law**:
For triangle \( AB_1D_1 \):
\[ \vec{B_1D_1} = \vec{D_1} - \vec{B_1} = k_3 \vec{d} - k_1 \vec{b} \]

5. **Express \( \vec{B_1C_1} \)**:
By the triangle law, we can express \( \vec{B_1C_1} \):
\[ \vec{B_1C_1} = \vec{C_1} - \vec{B_1} = k_2 \vec{c} - k_1 \vec{b} \]

6. **Establish Parallelism**:
Since \( \vec{B_1C_1} \) is parallel to \( \vec{B_1D_1} \), we can write:
\[ \vec{B_1C_1} = k \cdot \vec{B_1D_1} \quad \text{for some scalar } k \]

7. **Set Up the Equation**:
From the previous steps, we have:
\[ k_2 \vec{c} - k_1 \vec{b} = k \left( k_3 \vec{d} - k_1 \vec{b} \right) \]

8. **Compare Coefficients**:
Rearranging gives us two equations:
- Coefficient of \( \vec{b} \):
\[ -k_1 = -k k_1 \implies k = 1 \quad \text{(if } k_1 \neq 0\text{)} \]
- Coefficient of \( \vec{d} \):
\[ k_2 = k k_3 \implies k_2 = k_3 \quad \text{(if } k \neq 0\text{)} \]

9. **Final Relationships**:
From the above, we can derive:
\[ k_2 - k_1 = -k k_1 \quad \text{and} \quad k_2 = k k_3 \]

10. **Conclusion**:
We conclude that \( k_1, k_2, k_3 \) are in Harmonic Progression (HP) based on the derived relationships.