If `x = sqrt (1 + sqrt (1+ sqrt (1+ ...,)))` then x is equals

Options

A. `(1 + sqrt5)/( 2)`
B. `(2+ sqrt5)/( 2)`
C. `(-1 + sqrt5)/( 2)`
D. `(-1 - sqrt5)/( 2)`

Answer

To solve the equation \( x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \ldots}}} \), we can follow these steps:

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

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 1 + x \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ x^2 - x - 1 = 0 \]

### Step 4: Apply the quadratic formula
Now we will use the quadratic formula to solve for \( x \). The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation \( x^2 - x - 1 = 0 \), we have \( a = 1 \), \( b = -1 \), and \( c = -1 \).

Substituting these values into the quadratic formula:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 + 4}}{2} \]
\[ x = \frac{1 \pm \sqrt{5}}{2} \]

### Step 5: Determine the valid solution
Since \( x \) represents a length (as it is derived from a square root), we only consider the positive solution:
\[ x = \frac{1 + \sqrt{5}}{2} \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \frac{1 + \sqrt{5}}{2} \]

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