Write the next term of the A.P :
`3,3+sqrt2,3+2sqrt2,3+3sqrt2` …

Answer

To find the next term of the given arithmetic progression (A.P.):

**Given A.P.:**
3, 3 + √2, 3 + 2√2, 3 + 3√2

**Step 1: Identify the first term (a1) and the common difference (d).**
The first term (a1) is 3.
The second term (a2) is 3 + √2.
To find the common difference (d), we can subtract the first term from the second term:
\[ d = a2 - a1 = (3 + \sqrt{2}) - 3 = \sqrt{2} \]

**Step 2: Verify the common difference with the subsequent terms.**
Now, let's check the difference between the second and third terms:
\[ d = a3 - a2 = (3 + 2\sqrt{2}) - (3 + \sqrt{2}) = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]
And between the third and fourth terms:
\[ d = a4 - a3 = (3 + 3\sqrt{2}) - (3 + 2\sqrt{2}) = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]
So, the common difference is consistent at √2.

**Step 3: Calculate the next term (a5).**
To find the next term (a5), we add the common difference (d) to the fourth term (a4):
\[ a5 = a4 + d = (3 + 3\sqrt{2}) + \sqrt{2} \]
\[ a5 = 3 + 3\sqrt{2} + \sqrt{2} = 3 + 4\sqrt{2} \]

**Final Answer:**
The next term of the A.P. is \( 3 + 4\sqrt{2} \).

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