If `S_(r) = sqrt(r+ sqrt(r+sqrt(r+sqrt(......oo)))) , r gt 0` then which the following is\are correct.

Options

A. `S_(2), S_(6), S_(13), S _(20) ` are in A.P.
B. `S_(4), S_(9), S_(16)` are irrational
C. `(2S_(3)-1) ^(2), (2S_(4)-1)^(2) , (2S_(2)-1)^(2)` are in A.P.
D. `S_(2), S_(12), S_(36)` are in G.P.

Answer

To solve the problem \( S_r = \sqrt{r + \sqrt{r + \sqrt{r + \ldots}}} \) for \( r > 0 \), we can follow these steps:

### Step 1: Set up the equation
We start by recognizing that the expression inside the square root is the same as \( S_r \). Therefore, we can write:
\[ S_r = \sqrt{r + S_r} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ S_r^2 = r + S_r \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ S_r^2 - S_r - r = 0 \]

### Step 4: Apply the quadratic formula
Using the quadratic formula \( S_r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -1, c = -r \):
\[ S_r = \frac{1 \pm \sqrt{1 + 4r}}{2} \]

### Step 5: Choose the correct root
Since \( S_r \) must be positive (as \( r > 0 \)), we take the positive root:
\[ S_r = \frac{1 + \sqrt{1 + 4r}}{2} \]

### Step 6: Analyze the options
Now we need to check the options given in the question:

1. **First option**: Check if \( S_2, S_6, S_{12}, S_{20} \) are in arithmetic progression (AP).
- Calculate \( S_2, S_6, S_{12}, S_{20} \):
- \( S_2 = \frac{1 + \sqrt{1 + 8}}{2} = \frac{1 + 3}{2} = 2 \)
- \( S_6 = \frac{1 + \sqrt{1 + 24}}{2} = \frac{1 + 5}{2} = 3 \)
- \( S_{12} = \frac{1 + \sqrt{1 + 48}}{2} = \frac{1 + 7}{2} = 4 \)
- \( S_{20} = \frac{1 + \sqrt{1 + 80}}{2} = \frac{1 + 9}{2} = 5 \)
- The sequence \( 2, 3, 4, 5 \) has a common difference of 1, so it is in AP.

2. **Second option**: Check if \( S_4, S_9, S_{16} \) are irrational.
- Calculate \( S_4, S_9, S_{16} \):
- \( S_4 = \frac{1 + \sqrt{1 + 16}}{2} = \frac{1 + 5}{2} = 3 \) (irrational)
- \( S_9 = \frac{1 + \sqrt{1 + 36}}{2} = \frac{1 + 7}{2} = 4 \) (irrational)
- \( S_{16} = \frac{1 + \sqrt{1 + 64}}{2} = \frac{1 + 9}{2} = 5 \) (irrational)
- All values are irrational.

3. **Third option**: Check if \( 2S_2 - 1, 2S_4 - 1, 2S_6 - 1 \) are in AP.
- Calculate:
- \( 2S_2 - 1 = 2(2) - 1 = 3 \)
- \( 2S_4 - 1 = 2(3) - 1 = 5 \)
- \( 2S_6 - 1 = 2(4) - 1 = 7 \)
- The sequence \( 3, 5, 7 \) has a common difference of 2, so it is in AP.

4. **Fourth option**: Check if \( S_2, S_{12}, S_{56} \) are in geometric progression (GP).
- Calculate:
- \( S_{12} = 4 \) (as calculated before)
- \( S_{56} = \frac{1 + \sqrt{1 + 224}}{2} = \frac{1 + 15}{2} = 8 \)
- The sequence \( 2, 4, 8 \) has a common ratio of 2, so it is in GP.

### Conclusion
All options are correct.