By adding the same constant to each of 31,7,-1 a geometric progression results. The common ratio is

Options

A. 13
B. `2 1/3`
C. -12
D. none

Answer

To solve the problem, we need to determine the common ratio of a geometric progression (GP) formed by adding the same constant to each of the numbers 31, 7, and -1.

### Step-by-Step Solution:

1. **Define the constant to be added**: Let the constant we add to each number be \( x \). Therefore, the new numbers will be:
- First term: \( 31 + x \)
- Second term: \( 7 + x \)
- Third term: \( -1 + x \)

2. **Set up the condition for a geometric progression**: For three numbers \( a, b, c \) to be in GP, the condition is:
\[ \frac{b}{a} = \frac{c}{b} \]
Applying this to our terms, we have:
\[ \frac{7 + x}{31 + x} = \frac{-1 + x}{7 + x} \]

3. **Cross-multiply to eliminate the fractions**:
\[ (7 + x)(7 + x) = (31 + x)(-1 + x) \]

4. **Expand both sides**:
- Left side:
\[ (7 + x)^2 = 49 + 14x + x^2 \]
- Right side:
\[ (31 + x)(-1 + x) = -31 + 31x + x - x^2 = -31 + 32x - x^2 \]

5. **Set the two sides equal**:
\[ 49 + 14x + x^2 = -31 + 32x - x^2 \]

6. **Combine like terms**:
\[ 49 + 14x + x^2 + x^2 - 32x + 31 = 0 \]
This simplifies to:
\[ 2x^2 - 18x + 80 = 0 \]

7. **Divide the entire equation by 2**:
\[ x^2 - 9x + 40 = 0 \]

8. **Factor the quadratic equation**:
\[ (x - 5)(x - 8) = 0 \]
Thus, \( x = 5 \) or \( x = 8 \).

9. **Choose one value for \( x \)**: We can use either value, but let's use \( x = 5 \).

10. **Calculate the new terms**:
- First term: \( 31 + 5 = 36 \)
- Second term: \( 7 + 5 = 12 \)
- Third term: \( -1 + 5 = 4 \)

11. **Determine the common ratio**: The common ratio \( r \) can be calculated as:
\[ r = \frac{12}{36} = \frac{1}{3} \]

### Final Answer:
The common ratio of the geometric progression is \( \frac{1}{3} \).