Is the sequence `3,3 + sqrt(2), 3+ 2sqrt(2) , 3+ 3sqrt(2) , ....` an A.P. ? Justify.

Answer

To determine if the sequence \(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\) is an arithmetic progression (A.P.), we need to check if the common difference between successive terms is constant.

### Step-by-Step Solution:

1. **Identify the terms of the sequence**:
- Let \(a_1 = 3\)
- Let \(a_2 = 3 + \sqrt{2}\)
- Let \(a_3 = 3 + 2\sqrt{2}\)
- Let \(a_4 = 3 + 3\sqrt{2}\)

2. **Calculate the common difference between the first two terms**:
\[ d_1 = a_2 - a_1 = (3 + \sqrt{2}) - 3 = \sqrt{2} \]

3. **Calculate the common difference between the second and third terms**:
\[ d_2 = a_3 - a_2 = (3 + 2\sqrt{2}) - (3 + \sqrt{2}) = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]

4. **Calculate the common difference between the third and fourth terms**:
\[ d_3 = a_4 - a_3 = (3 + 3\sqrt{2}) - (3 + 2\sqrt{2}) = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]

5. **Conclusion**:
Since the common difference \(d_1 = d_2 = d_3 = \sqrt{2}\) is the same for all pairs of successive terms, we can conclude that the sequence is an arithmetic progression (A.P.).

### Final Answer:
Yes, the sequence \(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\) is an A.P. because the common difference between successive terms is constant and equal to \(\sqrt{2}\).

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