If the first and `(2n-1)^(th)` terms of an A.P, a G.P and an H.P of positive terms are equal and their `(n+1)^(th)` terms are `a, b` & `c` respectively then

Options

A. `a = b = c `
B. `a + c = 2b`
C. `a ge b ge c`
D. `ac = b^(2)`

Answer

To solve the problem step by step, we will analyze the given information about the A.P, G.P, and H.P.

### Step 1: Define the terms in the sequences
Let:
- The first term of the A.P be \( x \).
- The \( (2n-1)^{th} \) term of the A.P be \( y \).
- The first term of the G.P be \( x \).
- The \( (2n-1)^{th} \) term of the G.P be \( y \).
- The first term of the H.P be \( x \).
- The \( (2n-1)^{th} \) term of the H.P be \( y \).

Since the first and \( (2n-1)^{th} \) terms of the A.P, G.P, and H.P are equal, we have:
\[ x = y \]

### Step 2: Express the \( (n+1)^{th} \) terms
Now, we can express the \( (n+1)^{th} \) terms of each sequence:
- For the A.P:
\[ a = x + n \cdot d \quad \text{(where \( d \) is the common difference)} \]
- For the G.P:
\[ b = x \cdot r^n \quad \text{(where \( r \) is the common ratio)} \]
- For the H.P:
\[ c = \frac{2xy}{x + y} = \frac{2x^2}{2x} = x \quad \text{(since \( x = y \))} \]

### Step 3: Relate the terms
From the definitions, we know:
- The A.M (Arithmetic Mean) of \( x \) and \( y \) is:
\[ a = \frac{x + y}{2} = x \quad \text{(since \( x = y \))} \]
- The G.M (Geometric Mean) of \( x \) and \( y \) is:
\[ b = \sqrt{xy} = \sqrt{x^2} = x \]
- The H.M (Harmonic Mean) of \( x \) and \( y \) is:
\[ c = \frac{2xy}{x + y} = \frac{2x^2}{2x} = x \]

### Step 4: Establish relationships
From the above calculations, we can conclude:
\[ a = b = c = x \]

### Step 5: Verify inequalities
Using the properties of means:
- We know that:
\[ A.M \geq G.M \geq H.M \]
Thus:
\[ a \geq b \geq c \]

### Step 6: Apply the relationship between means
We also know from the relationship between the means:
\[ G.M^2 = A.M \cdot H.M \]
Substituting the values we found:
\[ b^2 = a \cdot c \]

### Conclusion
Thus, we have shown that:
1. \( a = b = c \)
2. \( a + c = 2b \)
3. \( a \geq b \geq c \)
4. \( ac = b^2 \)