`sqrt(6+sqrt(6+sqrt(6+sqrt(6+....................infty" times"))))=`

Options

A. 3
B. 2
C. 1
D. `pm3`

Answer

To solve the equation \( \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}}} \), we can follow these steps:

### Step 1: Define the expression
Let \( y \) be the value of the infinite nested square root:
\[ y = \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}}} \]

### Step 2: Set up the equation
Since the expression inside the square root is the same as \( y \), we can rewrite the equation as:
\[ y = \sqrt{6 + y} \]

### Step 3: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 6 + y \]

### Step 4: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ y^2 - y - 6 = 0 \]

### Step 5: Factor the quadratic equation
Next, we can factor the quadratic equation:
\[ (y - 3)(y + 2) = 0 \]

### Step 6: Solve for \( y \)
Setting each factor to zero gives us the possible solutions:
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]
\[ y + 2 = 0 \quad \Rightarrow \quad y = -2 \]

### Step 7: Determine the valid solution
Since \( y \) represents a length (the value of a square root), it must be non-negative. Therefore, we discard \( y = -2 \) and accept:
\[ y = 3 \]

### Final Answer
Thus, the value of the infinite nested square root is:
\[ \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \ldots}}}} = 3 \]

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