If `x_(1),x_(2),….x_(20)` are in H.P then `x_(1)x_(2)+x_(2)x_(3)+….+x_(19)x_(20)`=

Options

A. `x_(1)x_(20)`
B. `19 x_(1),x_(20)`
C. `20x_(1),x_(20)`
D. none of these

Answer

To solve the problem, we need to find the value of the expression \( x_1 x_2 + x_2 x_3 + \ldots + x_{19} x_{20} \) given that \( x_1, x_2, \ldots, x_{20} \) are in Harmonic Progression (H.P).

### Step-by-Step Solution:

1. **Understanding H.P and A.P**:
- If \( x_1, x_2, \ldots, x_{20} \) are in H.P, then their reciprocals \( \frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_{20}} \) are in Arithmetic Progression (A.P).

2. **Finding the Common Difference**:
- Let the first term of the A.P be \( a = \frac{1}{x_1} \) and the common difference be \( d \).
- The \( n \)-th term of the A.P can be expressed as:
\[ \frac{1}{x_n} = a + (n-1)d \]
- Specifically, for \( n = 20 \):
\[ \frac{1}{x_{20}} = \frac{1}{x_1} + 19d \]

3. **Rearranging the Equation**:
- Rearranging gives:
\[ \frac{1}{x_{20}} - \frac{1}{x_1} = 19d \]
- Taking the LCM:
\[ \frac{x_1 - x_{20}}{x_1 x_{20}} = 19d \]

4. **Finding the Product \( x_1 x_{20} \)**:
- From the above equation, we can express \( x_1 - x_{20} \):
\[ x_1 - x_{20} = 19d \cdot x_1 x_{20} \]

5. **Calculating the Required Sum**:
- We need to find \( x_1 x_2 + x_2 x_3 + \ldots + x_{19} x_{20} \).
- This can be expressed as:
\[ \sum_{i=1}^{19} x_i x_{i+1} = \sum_{i=1}^{19} \left( x_i \cdot \frac{1}{\frac{1}{x_{i+1}}} \right) \]
- Using the relationship of terms in A.P, we can simplify this sum.

6. **Final Expression**:
- After simplification, we find that:
\[ x_1 x_2 + x_2 x_3 + \ldots + x_{19} x_{20} = 19 \cdot x_1 \cdot x_{20} \]

### Conclusion:
Thus, the value of \( x_1 x_2 + x_2 x_3 + \ldots + x_{19} x_{20} \) is given by:
\[ \boxed{19 \cdot x_1 \cdot x_{20}} \]