`y=sqrt(5sqrt(5sqrt(5.....oo)))`, then value of is?

Options

A. 4
B. `2^((n-1)/(2n))`
C. 5
D. None of these

Answer

To solve the equation \( y = \sqrt{5\sqrt{5\sqrt{5\ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
We start with the equation:
\[ y = \sqrt{5\sqrt{5\sqrt{5\ldots}}} \]
Since the expression inside the square root is the same as \( y \), we can rewrite the equation as:
\[ y = \sqrt{5y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 5y \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ y^2 - 5y = 0 \]

### Step 4: Factor the equation
We can factor the left-hand side:
\[ y(y - 5) = 0 \]

### Step 5: Solve for \( y \)
Setting each factor equal to zero gives us:
1. \( y = 0 \)
2. \( y - 5 = 0 \) which simplifies to \( y = 5 \)

### Step 6: Determine the valid solution
Since \( y \) represents a square root and must be non-negative, we discard \( y = 0 \). Thus, the valid solution is:
\[ y = 5 \]

### Final Answer
The value of \( y \) is \( 5 \).

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