Determine the value of 'z' in the equation :
`z=sqrt(20+sqrt(20+sqrt(20+sqrt(20+...oo))))`

Options

A. 4
B. 5
C. 20
D. 25

Answer

To solve the equation \( z = \sqrt{20 + \sqrt{20 + \sqrt{20 + \sqrt{20 + \ldots}}}} \), we can follow these steps:

### Step 1: Set up the equation
We start with the given equation:
\[ z = \sqrt{20 + \sqrt{20 + \sqrt{20 + \ldots}}} \]
Since the expression inside the square root is the same as \( z \), we can rewrite the equation as:
\[ z = \sqrt{20 + z} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ z^2 = 20 + z \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to form a standard quadratic equation:
\[ z^2 - z - 20 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation. We look for two numbers that multiply to \(-20\) and add to \(-1\). The numbers \(-5\) and \(4\) work:
\[ (z - 5)(z + 4) = 0 \]

### Step 5: Solve for \( z \)
Setting each factor to zero gives us the possible solutions:
\[ z - 5 = 0 \quad \Rightarrow \quad z = 5 \]
\[ z + 4 = 0 \quad \Rightarrow \quad z = -4 \]

### Step 6: Determine the valid solution
Since \( z \) represents a square root, it must be non-negative. Therefore, we discard \( z = -4 \) and keep:
\[ z = 5 \]

### Final Answer
The value of \( z \) is:
\[ \boxed{5} \]