The 1st, 8th and 22nd terms of an AP are three conscutive terms of a GP. Find the common ratio of the GP, given that the sum of the first twenty-two terms of the AP is 385.

Options

A. either 1 or 1/2
B. 2
C. 1
D. either 1 or 3

Answer

To solve the problem, we need to find the common ratio of the geometric progression (GP) formed by the 1st, 8th, and 22nd terms of an arithmetic progression (AP), given that the sum of the first 22 terms of the AP is 385.

### Step-by-Step Solution:

1. **Define the Terms of the AP**:
- Let the first term of the AP be \( A \) and the common difference be \( D \).
- The first term (1st term) of the AP is \( A \).
- The eighth term (8th term) of the AP is \( A + 7D \).
- The twenty-second term (22nd term) of the AP is \( A + 21D \).

2. **Set Up the GP Condition**:
- The terms \( A \), \( A + 7D \), and \( A + 21D \) are in GP.
- For three terms to be in GP, the square of the middle term must equal the product of the other two terms:
\[ (A + 7D)^2 = A \cdot (A + 21D) \]

3. **Expand and Rearrange**:
- Expanding both sides:
\[ A^2 + 14AD + 49D^2 = A^2 + 21AD \]
- Rearranging gives:
\[ 14AD + 49D^2 = 21AD \]
- Simplifying leads to:
\[ 49D^2 = 7AD \quad \Rightarrow \quad 7D(7D - A) = 0 \]
- This implies either \( D = 0 \) (which is not possible since it would mean no progression) or \( A = 7D \).

4. **Use the Sum of the First 22 Terms of the AP**:
- The sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]
- For \( n = 22 \):
\[ S_{22} = \frac{22}{2} \times (2A + 21D) = 11(2A + 21D) \]
- We know \( S_{22} = 385 \):
\[ 11(2A + 21D) = 385 \quad \Rightarrow \quad 2A + 21D = \frac{385}{11} = 35 \]

5. **Substitute \( A = 7D \) into the Sum Equation**:
- Substitute \( A = 7D \):
\[ 2(7D) + 21D = 35 \quad \Rightarrow \quad 14D + 21D = 35 \quad \Rightarrow \quad 35D = 35 \]
- Therefore, \( D = 1 \).

6. **Find \( A \)**:
- Since \( D = 1 \):
\[ A = 7D = 7 \times 1 = 7 \]

7. **Calculate the Common Ratio \( R \)**:
- The common ratio \( R \) of the GP is given by:
\[ R = \frac{A + 7D}{A} = \frac{7 + 7 \cdot 1}{7} = \frac{14}{7} = 2 \]

### Final Answer:
The common ratio of the GP is \( R = 2 \).