The first terms of a G.P. is 2 and sum to infinity is 6, find common ratio.

Answer

To solve the problem, we need to find the common ratio \( r \) of a geometric progression (G.P.) given that the first term \( a \) is 2 and the sum to infinity \( S \) is 6.

### Step-by-step Solution:

1. **Identify the first term and the sum to infinity**:
- Given: \( a = 2 \) (the first term of the G.P.)
- Given: \( S = 6 \) (the sum to infinity)

2. **Use the formula for the sum to infinity of a G.P.**:
- The formula for the sum to infinity of a G.P. is:
\[ S = \frac{a}{1 - r} \]
- Substituting the known values into the formula:
\[ 6 = \frac{2}{1 - r} \]

3. **Cross-multiply to eliminate the fraction**:
- Rearranging the equation gives:
\[ 6(1 - r) = 2 \]

4. **Distribute the 6**:
- Expanding the left side:
\[ 6 - 6r = 2 \]

5. **Isolate the term involving \( r \)**:
- Rearranging the equation to isolate \( r \):
\[ 6 - 2 = 6r \]
- This simplifies to:
\[ 4 = 6r \]

6. **Solve for \( r \)**:
- Dividing both sides by 6:
\[ r = \frac{4}{6} \]
- Simplifying the fraction:
\[ r = \frac{2}{3} \]

7. **Conclusion**:
- The common ratio \( r \) of the G.P. is \( \frac{2}{3} \).