Find the value of `sqrt(1+2sqrt(1+2sqrt(1+2sqrt(1+......))))`

Answer

To find the value of \( \sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + \ldots}}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( x = \sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + \ldots}}}} \). This means we can express \( x \) as:
\[ x = \sqrt{1 + 2x} \]

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

### Step 3: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:
\[ x^2 - 2x - 1 = 0 \]

### Step 4: Use the quadratic formula
We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -2 \), and \( c = -1 \).

### Step 5: Substitute the values into the formula
Substituting the values into the formula:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]

### Step 6: Simplify the square root
Since \( \sqrt{8} = 2\sqrt{2} \), we can write:
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
This simplifies to:
\[ x = 1 \pm \sqrt{2} \]

### Step 7: Determine the valid solution
Now we have two potential solutions:
1. \( x = 1 + \sqrt{2} \)
2. \( x = 1 - \sqrt{2} \)

Since \( \sqrt{2} \) is approximately \( 1.414 \), the second solution \( 1 - \sqrt{2} \) is negative, which is not valid in this context because \( x \) must be non-negative. Therefore, we take:
\[ x = 1 + \sqrt{2} \]

### Final Answer
Thus, the value of \( \sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + 2\sqrt{1 + \ldots}}}} \) is:
\[ \boxed{1 + \sqrt{2}} \]