If `p^("th"), 2p^("th") and 4p^("th")` terms of an arithmetic progression are in geometric progression, then the common ratio of the geometric progression is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to determine the common ratio of the geometric progression formed by the terms \( p^{th}, 2p^{th}, \) and \( 4p^{th} \) of an arithmetic progression.

### Step-by-Step Solution:

1. **Identify the Terms in AP**:
Let the first term of the arithmetic progression (AP) be \( a \) and the common difference be \( d \).
- The \( p^{th} \) term, \( T_p \), is given by:
\[ T_p = a + (p - 1)d \]
- The \( 2p^{th} \) term, \( T_{2p} \), is given by:
\[ T_{2p} = a + (2p - 1)d \]
- The \( 4p^{th} \) term, \( T_{4p} \), is given by:
\[ T_{4p} = a + (4p - 1)d \]

2. **Condition for GP**:
The terms \( T_p, T_{2p}, T_{4p} \) are in geometric progression (GP) if:
\[ T_{2p}^2 = T_p \cdot T_{4p} \]

3. **Substituting the Terms**:
Substitute the expressions for \( T_p, T_{2p}, T_{4p} \) into the GP condition:
\[ (a + (2p - 1)d)^2 = (a + (p - 1)d)(a + (4p - 1)d) \]

4. **Expanding Both Sides**:
- Left-hand side:
\[ (a + (2p - 1)d)^2 = a^2 + 2a(2p - 1)d + (2p - 1)^2d^2 \]
- Right-hand side:
\[ (a + (p - 1)d)(a + (4p - 1)d = a^2 + (4p - 1 + p - 1)ad + (p - 1)(4p - 1)d^2 \]
\[ = a^2 + (5p - 2)ad + (4p^2 - 5p + 1)d^2 \]

5. **Setting the Equations Equal**:
Equate the left-hand side and right-hand side:
\[ a^2 + 2a(2p - 1)d + (2p - 1)^2d^2 = a^2 + (5p - 2)ad + (4p^2 - 5p + 1)d^2 \]

6. **Cancelling \( a^2 \)**:
Cancel \( a^2 \) from both sides:
\[ 2a(2p - 1)d + (2p - 1)^2d^2 = (5p - 2)ad + (4p^2 - 5p + 1)d^2 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ (2(2p - 1) - (4p^2 - 5p + 1))d^2 + (2a(2p - 1) - (5p - 2))ad = 0 \]

8. **Finding the Common Ratio**:
The common ratio \( r \) of the GP is given by:
\[ r = \frac{T_{2p}}{T_p} = \frac{a + (2p - 1)d}{a + (p - 1)d} \]
Substituting \( a = d \) (from earlier simplifications), we have:
\[ r = \frac{d + (2p - 1)d}{d + (p - 1)d} = \frac{(2p)d}{(p)d} = 2 \]

### Conclusion:
The common ratio of the geometric progression is \( \boxed{2} \).