If the pth ,qth and rth terms of an AP are in G.P then the common ration of the GP is

Options

A. `pr/q^2`
B. `r/p`
C. `(q+r)/(p+q)`
D. `(q-r)/(p-q)`

Explanation

To find the common ratio of the GP formed by the pth, qth, and rth terms of an AP, we will follow these steps:

### Step 1: Define the terms of the AP
The pth, qth, and rth terms of an arithmetic progression (AP) can be expressed as:
- \( T_p = a + (p - 1)d \)
- \( T_q = a + (q - 1)d \)
- \( T_r = a + (r - 1)d \)

Where:
- \( a \) is the first term of the AP,
- \( d \) is the common difference.

### Step 2: Set up the relationship for GP
Since \( T_p, T_q, T_r \) are in a geometric progression (GP), we can use the property of GP:
\[ T_q^2 = T_p \cdot T_r \]

### Step 3: Substitute the terms into the GP condition
Substituting the expressions for \( T_p, T_q, T_r \):
\[ \left( a + (q - 1)d \right)^2 = \left( a + (p - 1)d \right) \cdot \left( a + (r - 1)d \right) \]

### Step 4: Expand both sides
Expanding both sides gives:
\[ \left( a^2 + 2a(q - 1)d + (q - 1)^2d^2 \right) = \left( a^2 + (p + r - 2)a d + (p - 1)(r - 1)d^2 \right) \]

### Step 5: Simplify the equation
By simplifying both sides, we can cancel \( a^2 \) from both sides:
\[ 2a(q - 1)d + (q - 1)^2d^2 = (p + r - 2)ad + (p - 1)(r - 1)d^2 \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 2a(q - 1)d - (p + r - 2)ad = (p - 1)(r - 1)d^2 - (q - 1)^2d^2 \]

### Step 7: Factor out common terms
Factoring out \( d \) from both sides:
\[ d \left( 2a(q - 1) - (p + r - 2)a \right) = d^2 \left( (p - 1)(r - 1) - (q - 1)^2 \right) \]

### Step 8: Divide by \( d \) (assuming \( d \neq 0 \))
Dividing both sides by \( d \) gives:
\[ 2a(q - 1) - (p + r - 2)a = d \left( (p - 1)(r - 1) - (q - 1)^2 \right) \]

### Step 9: Solve for the common ratio
From the GP property, we can find the common ratio \( r \) as:
\[ r = \frac{T_q}{T_p} = \frac{a + (q - 1)d}{a + (p - 1)d} \]
And similarly,
\[ r = \frac{T_r}{T_q} = \frac{a + (r - 1)d}{a + (q - 1)d} \]

### Step 10: Final expression for the common ratio
By manipulating these ratios, we find that:
\[ r = \frac{q - r}{p - q} \]

### Final Answer
The common ratio of the GP formed by the pth, qth, and rth terms of the AP is:
\[ \frac{q - r}{p - q} \]