`sqrt(1+sqrt(1+sqrt(1+…..)))`

Options

A. equals 1
B. lies between 0 and 1
C. lies between 1 and 2
D. is greater than 2

Answer

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

### Step 1: Define the Expression
Let \( x \) be the value of the entire expression:
\[ x = \sqrt{1 + \sqrt{1 + \sqrt{1 + \ldots}}} \]
This means that:
\[ x = \sqrt{1 + x} \]

### 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
Rearranging the equation gives us:
\[ x^2 - x - 1 = 0 \]

### Step 4: Apply the Quadratic Formula
We can solve the quadratic equation \( ax^2 + bx + c = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -1 \). Plugging in these values:
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \]

### Step 5: Determine the Values of \( x \)
Now we have two potential solutions for \( x \):
\[ x_1 = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad x_2 = \frac{1 - \sqrt{5}}{2} \]
Calculating \( \sqrt{5} \) gives approximately \( 2.236 \). Therefore:
- \( x_1 \approx \frac{1 + 2.236}{2} = \frac{3.236}{2} \approx 1.618 \)
- \( x_2 \approx \frac{1 - 2.236}{2} = \frac{-1.236}{2} \approx -0.618 \)

### Step 6: Select the Positive Solution
Since \( x \) represents a length (as it is derived from square roots), we discard the negative solution:
\[ x = \frac{1 + \sqrt{5}}{2} \approx 1.618 \]

### Conclusion
The value of \( \sqrt{1 + \sqrt{1 + \sqrt{1 + \ldots}}} \) is approximately \( 1.618 \), which lies between 1 and 2. Therefore, the correct option is:
**Option 3: Between 1 and 2.**