If `4=sqrt(x + sqrt(x + sqrt(x +..)))`, then the value of x will be:

Options

A. 20
B. 16
C. 12
D. 8

Answer

To solve the equation \( 4 = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
We start with the given equation:
\[ 4 = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} \]
Let \( y = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} \). Thus, we can rewrite the equation as:
\[ y = \sqrt{x + y} \]

### Step 2: Substitute the value of \( y \)
Since we know \( y = 4 \), we substitute this into the equation:
\[ 4 = \sqrt{x + 4} \]

### Step 3: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ 4^2 = (\sqrt{x + 4})^2 \]
This simplifies to:
\[ 16 = x + 4 \]

### Step 4: Solve for \( x \)
Now, we isolate \( x \) by subtracting 4 from both sides:
\[ 16 - 4 = x \]
Thus, we find:
\[ x = 12 \]

### Final Answer
The value of \( x \) is:
\[ \boxed{12} \]