Find the 5th term of the progression `1,((sqrt(2)-1))/(2sqrt(3)),\ ((3-2sqrt(2))/(12)),\ ((5sqrt(2)-7)/(24sqrt(3))),\ ddot`

Explanation

To find the 5th term of the given progression \(1, \frac{\sqrt{2}-1}{2\sqrt{3}}, \frac{3-2\sqrt{2}}{12}, \frac{5\sqrt{2}-7}{24\sqrt{3}}, \ldots\), we will first determine if this sequence is a geometric progression (GP) by calculating the ratios of consecutive terms. If the ratios are constant, we can use the formula for the nth term of a GP.

### Step 1: Identify the terms
Let:
- \(A_1 = 1\)
- \(A_2 = \frac{\sqrt{2}-1}{2\sqrt{3}}\)
- \(A_3 = \frac{3-2\sqrt{2}}{12}\)
- \(A_4 = \frac{5\sqrt{2}-7}{24\sqrt{3}}\)
...