The value of `sqrt(7+sqrt(7-sqrt(7+sqrt(7-….))))` upto `oo` is

Options

A. 5
B. 4
C. 3
D. 2

Explanation

To solve the equation \( x = \sqrt{7 + \sqrt{7 - \sqrt{7 + \sqrt{7 - \ldots}}}} \), we will follow these steps:

### Step 1: Set up the equation
Let \( x = \sqrt{7 + \sqrt{7 - x}} \).

### Step 2: Square both sides
Squaring both sides gives:
\[ x^2 = 7 + \sqrt{7 - x} \]

### Step 3: Isolate the square root
Rearranging the equation, we have:
\[ \sqrt{7 - x} = x^2 - 7 \]

### Step 4: Square again
Now, square both sides again:
\[ 7 - x = (x^2 - 7)^2 \]

### Step 5: Expand the right side
Expanding the right side:
\[ 7 - x = x^4 - 14x^2 + 49 \]

### Step 6: Rearrange to form a polynomial equation
Rearranging gives:
\[ x^4 - 14x^2 + x + 42 = 0 \]

### Step 7: Test possible rational roots
We will test possible rational roots. Let's try \( x = 3 \):
\[ 3^4 - 14(3^2) + 3 + 42 = 81 - 126 + 3 + 42 = 0 \]
Thus, \( x = 3 \) is a root.

### Conclusion
The value of \( \sqrt{7 + \sqrt{7 - \sqrt{7 + \sqrt{7 - \ldots}}}} \) is \( \boxed{3} \).

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