If `a, a_1 ,a_2----a_(2n-1),b` are in `A.P and a,b_1,b_2-----b_(2n-1),b` are in `G.P and a,c_1,c_2----c_(2n-1),b` are in `H.P` (which are non-zero and a,b are positive real numbers), then the roots of the equation `a_nx^2-b_nx +c_n=0` are

Options

A. `a_(n)^(2)=b_(n)c_(n)`
B. `b_(n)^(2)=c_(n)a_(n)`
C. `c_(n)^(2)=a_(n)b_(n)`
D. none of these

Explanation

To solve the problem step by step, we need to analyze the given sequences and derive the necessary values for the quadratic equation \( a_n x^2 - b_n x + c_n = 0 \).

### Step 1: Understand the sequences
We are given three sequences:
1. \( a, a_1, a_2, \ldots, a_{2n-1}, b \) is in Arithmetic Progression (A.P).
2. \( a, b_1, b_2, \ldots, b_{2n-1}, b \) is in Geometric Progression (G.P).
3. \( a, c_1, c_2, \ldots, c_{2n-1}, b \) is in Harmonic Progression (H.P).

### Step 2: Find \( a_n \) from A.P
In an A.P, the \( n \)-th term \( a_n \) can be calculated as:
\[ a_n = \frac{a + b}{2} \]
This is because the middle term of an A.P with \( 2n + 1 \) terms is the average of the first and last terms.

### Step 3: Find \( b_n \) from G.P
In a G.P, the \( n \)-th term \( b_n \) can be calculated as:
\[ b_n = \sqrt{ab} \]
This is because the middle term of a G.P with \( 2n + 1 \) terms is the geometric mean of the first and last terms.

### Step 4: Find \( c_n \) from H.P
In an H.P, the \( n \)-th term \( c_n \) can be calculated as:
\[ c_n = \frac{2ab}{a + b} \]
This is derived from the relationship of the H.P, where the harmonic mean is defined as the reciprocal of the average of the reciprocals.

### Step 5: Formulate the discriminant
The quadratic equation is given by:
\[ a_n x^2 - b_n x + c_n = 0 \]
To determine the nature of the roots, we need to calculate the discriminant \( D \):
\[ D = b_n^2 - 4a_n c_n \]

### Step 6: Substitute values into the discriminant
Substituting the values we found:
1. \( b_n = \sqrt{ab} \) gives \( b_n^2 = ab \).
2. \( a_n = \frac{a + b}{2} \).
3. \( c_n = \frac{2ab}{a + b} \).

Now substituting these into the discriminant:
\[ D = ab - 4 \left(\frac{a + b}{2}\right) \left(\frac{2ab}{a + b}\right) \]
Simplifying this:
\[ D = ab - 4ab = -3ab \]

### Step 7: Analyze the discriminant
Since \( a \) and \( b \) are positive real numbers, \( ab > 0 \). Therefore:
\[ D = -3ab < 0 \]

### Conclusion
When the discriminant is less than zero, it indicates that the roots of the quadratic equation are non-real (imaginary).

Thus, the roots of the equation \( a_n x^2 - b_n x + c_n = 0 \) are **non-real**.