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 \( x = \sqrt{7 + \sqrt{7 - \sqrt{7 + \sqrt{7 - \ldots}}}} \) up to infinity, we can follow these steps:

### Step 1: Set up the equation
Let \( x = \sqrt{7 + \sqrt{7 - \sqrt{7 + \sqrt{7 - \ldots}}}} \). This implies that the expression inside the square root is also equal to \( x \).

### Step 2: Rewrite the equation
From the definition, we can rewrite the equation as:
\[ x = \sqrt{7 + \sqrt{7 - x}} \]

### Step 3: Square both sides
To eliminate the square root, we square both sides:
\[ x^2 = 7 + \sqrt{7 - x} \]

### Step 4: Isolate the square root
Rearranging gives us:
\[ \sqrt{7 - x} = x^2 - 7 \]

### Step 5: Square again
We square both sides again to eliminate the remaining square root:
\[ 7 - x = (x^2 - 7)^2 \]

### Step 6: Expand and rearrange
Expanding the right side:
\[ 7 - x = x^4 - 14x^2 + 49 \]
Rearranging gives us:
\[ x^4 - 14x^2 + x + 42 = 0 \]

### Step 7: Factor the polynomial
We can try to find the roots of the polynomial. Testing \( x = 3 \) and \( x = -2 \):
\[ (3)^4 - 14(3)^2 + 3 + 42 = 81 - 126 + 3 + 42 = 0 \]
Thus, \( x = 3 \) is a root. We can factor the polynomial as:
\[ (x - 3)(x + 2)(x^2 + x + 7) = 0 \]

### Step 8: Solve for remaining roots
The quadratic \( x^2 + x + 7 = 0 \) has no real roots (discriminant is negative). Thus, the only real solutions are \( x = 3 \) and \( x = -2 \).

### Step 9: Determine the valid solution
Since we are looking for a positive solution (as the original expression involves square roots), we discard \( x = -2 \). Therefore, the value of \( x \) is:
\[ \boxed{3} \]