Find which of the following is a G.P. :
`2,2sqrt(2),4,4sqrt(2)`, . . . . . . . . . . . . .

Answer

To determine if the sequence \(2, 2\sqrt{2}, 4, 4\sqrt{2}\) 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 of the sequence:**
- Let \(a_1 = 2\)
- Let \(a_2 = 2\sqrt{2}\)
- Let \(a_3 = 4\)
- Let \(a_4 = 4\sqrt{2}\)

2. **Calculate the ratio of the first two terms:**
\[ r_1 = \frac{a_2}{a_1} = \frac{2\sqrt{2}}{2} = \sqrt{2} \]

3. **Calculate the ratio of the second and third terms:**
\[ r_2 = \frac{a_3}{a_2} = \frac{4}{2\sqrt{2}} = \frac{4}{2\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{4} = \sqrt{2} \]

4. **Calculate the ratio of the third and fourth terms:**
\[ r_3 = \frac{a_4}{a_3} = \frac{4\sqrt{2}}{4} = \sqrt{2} \]

5. **Check if all ratios are equal:**
- We found that \(r_1 = \sqrt{2}\), \(r_2 = \sqrt{2}\), and \(r_3 = \sqrt{2}\).
- Since \(r_1 = r_2 = r_3\), the ratio is constant.

6. **Conclusion:**
- Since the ratio of consecutive terms is constant, the sequence \(2, 2\sqrt{2}, 4, 4\sqrt{2}\) is a geometric progression (G.P.).