The value of ${\sqrt{4+\sqrt{4+\sqrt{4+..........}}}}$ is :

Question

Accepted Answer

To solve the problem of finding the value of the expression $\sqrt{4+\sqrt{4+\sqrt{4+\ldots}}}$, we need to recognize that this is an infinite nested radical. Let's denote the entire expression by $x$. Therefore, we have:

$$ x = \sqrt{4 + \sqrt{4 + \sqrt{4 + \ldots}}} $$

This implies:

$$ x = \sqrt{4 + x} $$

To eliminate the square root, we square both sides:

$$ x^2 = 4 + x $$

Rearranging the equation gives us a quadratic equation:

$$ x^2 - x - 4 = 0 $$

To solve this quadratic equation, we use the quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

where $a = 1$, $b = -1$, and $c = -4$. Plugging these values into the formula gives:

$$ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} $$

$$ x = \frac{1 \pm \sqrt{1 + 16}}{2} $$

$$ x = \frac{1 \pm \sqrt{17}}{2} $$

This results in two potential solutions:

$$ x = \frac{1 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{17}}{2} $$

Since $x$ represents a length (the value of a nested radical), it must be positive. Therefore, the negative solution $\frac{1 - \sqrt{17}}{2}$ is not valid in this context. Thus, the correct solution is:

$$ x = \frac{1 + \sqrt{17}}{2} $$

This corresponds to Option 3.

**Explanation of why other options are incorrect:**

1. **Option 1 (1):** If $x = 1$, substituting into the equation $x = \sqrt{4 + x}$ gives $1 = \sqrt{4 + 1}$, which simplifies to $1 = \sqrt{5}$. This is not true, so Option 1 is incorrect.

2. **Option 2 $\left(\frac{1-\sqrt{17}}{2}\right)$:** As explained, this solution is negative, which is not possible for a nested radical expression that represents a length or magnitude. Therefore, Option 2 is incorrect.

3. **Option 4 (4):** If $x = 4$, substituting into the equation $x = \sqrt{4 + x}$ gives $4 = \sqrt{4 + 4}$, which simplifies to $4 = \sqrt{8}$. This is not true, so Option 4 is incorrect.

Thus, the only valid and correct option is Option 3: $\frac{1+\sqrt{17}}{2}$.