The ratio of the sum of 3 terms to the sum of 6 terms of a G.P. is 125:152. Its common ratio is :

Options

A. `(3)/(5)`
B. `(2)/(5)`
C. `(1)/(5)`
D. 2

Explanation

To solve the problem, we need to find the common ratio \( r \) of a geometric progression (G.P.) given that the ratio of the sum of the first 3 terms to the sum of the first 6 terms is \( \frac{125}{152} \).

### Step-by-Step Solution:

1. **Understanding the Sums of a G.P.**:
The sum of the first \( n \) terms of a G.P. can be expressed as:
\[ S_n = A \frac{r^n - 1}{r - 1} \]
where \( A \) is the first term and \( r \) is the common ratio.

2. **Setting Up the Equation**:
For \( n = 3 \) (sum of the first 3 terms):
\[ S_3 = A \frac{r^3 - 1}{r - 1} \]
For \( n = 6 \) (sum of the first 6 terms):
\[ S_6 = A \frac{r^6 - 1}{r - 1} \]
According to the problem, we have:
\[ \frac{S_3}{S_6} = \frac{125}{152} \]

3. **Substituting the Sums**:
Substituting the expressions for \( S_3 \) and \( S_6 \) into the ratio:
\[ \frac{A \frac{r^3 - 1}{r - 1}}{A \frac{r^6 - 1}{r - 1}} = \frac{125}{152} \]
The \( A \) and \( (r - 1) \) terms cancel out:
\[ \frac{r^3 - 1}{r^6 - 1} = \frac{125}{152} \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 152(r^3 - 1) = 125(r^6 - 1) \]
Expanding both sides:
\[ 152r^3 - 152 = 125r^6 - 125 \]

5. **Rearranging the Equation**:
Rearranging the equation leads to:
\[ 125r^6 - 152r^3 - 73 = 0 \]

6. **Letting \( x = r^3 \)**:
Let \( x = r^3 \), then the equation becomes:
\[ 125x^2 - 152x - 73 = 0 \]

7. **Using the Quadratic Formula**:
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 125 \), \( b = -152 \), and \( c = -73 \):
\[ x = \frac{152 \pm \sqrt{(-152)^2 - 4 \cdot 125 \cdot (-73)}}{2 \cdot 125} \]

8. **Calculating the Discriminant**:
Calculate the discriminant:
\[ (-152)^2 = 23104 \]
\[ 4 \cdot 125 \cdot 73 = 36500 \]
Thus,
\[ b^2 - 4ac = 23104 + 36500 = 59604 \]

9. **Finding the Roots**:
Now substituting back into the formula:
\[ x = \frac{152 \pm \sqrt{59604}}{250} \]
Calculate \( \sqrt{59604} \):
\[ \sqrt{59604} \approx 244 \]
Thus,
\[ x = \frac{152 \pm 244}{250} \]

10. **Calculating the Values**:
This gives two possible values for \( x \):
\[ x_1 = \frac{396}{250} = 1.584 \quad \text{and} \quad x_2 = \frac{-92}{250} \quad (\text{not valid since } r^3 > 0) \]

11. **Finding \( r \)**:
Since \( x = r^3 \), we take:
\[ r^3 = 1.584 \implies r = (1.584)^{1/3} \approx 1.174 \]

### Final Answer:
The common ratio \( r \) is approximately \( 1.174 \).