If `a_(1),a_(2),a_(3),a_(4),a_(5)` are in HP, then `a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5)` is equal to

Options

A. `2a_(1)a_(5)`
B. `3a_(1)a_(5)`
C. `4a_(1)a_(5)`
D. `6a_(1)a_(5)`

Explanation

To solve the problem, we need to find the value of the expression \( a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 \) given that \( a_1, a_2, a_3, a_4, a_5 \) are in Harmonic Progression (HP).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
- If \( a_1, a_2, a_3, a_4, a_5 \) are in HP, then their reciprocals \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \frac{1}{a_4}, \frac{1}{a_5} \) are in Arithmetic Progression (AP).

2. **Expressing Terms in AP**:
- Let \( \frac{1}{a_1} = x, \frac{1}{a_2} = x + d, \frac{1}{a_3} = x + 2d, \frac{1}{a_4} = x + 3d, \frac{1}{a_5} = x + 4d \) for some common difference \( d \).

3. **Finding the Products**:
- We need to compute \( a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 \).
- Using the expressions for \( a_i \):
\[ a_1 = \frac{1}{x}, \quad a_2 = \frac{1}{x + d}, \quad a_3 = \frac{1}{x + 2d}, \quad a_4 = \frac{1}{x + 3d}, \quad a_5 = \frac{1}{x + 4d} \]

4. **Calculating Each Product**:
- Calculate each product:
\[ a_1 a_2 = \frac{1}{x} \cdot \frac{1}{x + d} = \frac{1}{x(x + d)} \]
\[ a_2 a_3 = \frac{1}{x + d} \cdot \frac{1}{x + 2d} = \frac{1}{(x + d)(x + 2d)} \]
\[ a_3 a_4 = \frac{1}{x + 2d} \cdot \frac{1}{x + 3d} = \frac{1}{(x + 2d)(x + 3d)} \]
\[ a_4 a_5 = \frac{1}{x + 3d} \cdot \frac{1}{x + 4d} = \frac{1}{(x + 3d)(x + 4d)} \]

5. **Summing Up the Products**:
- Now we need to sum these products:
\[ S = a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 \]
- This can be simplified using the properties of fractions, but a more straightforward approach is to recognize a pattern in the sums of products of terms in HP.

6. **Using Known Result**:
- From properties of HP, we can derive that:
\[ a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 = 4 a_1 a_5 \]

7. **Final Result**:
- Thus, the expression \( a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 \) simplifies to \( 4 a_1 a_5 \).

### Conclusion:
The value of \( a_1 a_2 + a_2 a_3 + a_3 a_4 + a_4 a_5 \) is \( 4 a_1 a_5 \).