Find which of the following is a G.P. :
`xy,x^(2)y,x^(3)y`, . . . . . . . . . . . . .

Answer

To determine if the sequence \( xy, x^2y, x^3y, \ldots \) is a geometric progression (G.P.), we need to check if the ratio of consecutive terms is constant.

### Step-by-Step Solution:

1. **Identify the terms**:
The given terms are:
- \( T_1 = xy \)
- \( T_2 = x^2y \)
- \( T_3 = x^3y \)

2. **Calculate the ratio of the second term to the first term**:
\[ \text{Ratio} = \frac{T_2}{T_1} = \frac{x^2y}{xy} \]
Simplifying this:
\[ = \frac{x^2}{x} = x \]

3. **Calculate the ratio of the third term to the second term**:
\[ \text{Ratio} = \frac{T_3}{T_2} = \frac{x^3y}{x^2y} \]
Simplifying this:
\[ = \frac{x^3}{x^2} = x \]

4. **Compare the ratios**:
Since both ratios are equal:
\[ \frac{T_2}{T_1} = \frac{T_3}{T_2} = x \]
This indicates that the ratio of consecutive terms is constant.

5. **Conclusion**:
Since the ratio of consecutive terms is constant, the sequence \( xy, x^2y, x^3y, \ldots \) is a geometric progression (G.P.).