Find the value of `sqrt(30+sqrt(30+sqrt(30+……)))`

Options

A. `5`
B. `3sqrt(10)`
C. `6`
D. `7`

Answer

To solve the expression \( \sqrt{30 + \sqrt{30 + \sqrt{30 + \ldots}}} \), we can follow these steps:

### Step 1: Set up the equation
Let \( y = \sqrt{30 + \sqrt{30 + \sqrt{30 + \ldots}}} \). This implies that \( y \) can be expressed as:
\[ y = \sqrt{30 + y} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ y^2 = 30 + y \]

### Step 3: Rearrange the equation
Rearranging the equation gives us a standard quadratic form:
\[ y^2 - y - 30 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \(-30\) and add to \(-1\). The numbers \(-6\) and \(5\) satisfy this condition:
\[ (y - 6)(y + 5) = 0 \]

### Step 5: Solve for \( y \)
Setting each factor to zero gives us the possible solutions:
1. \( y - 6 = 0 \) → \( y = 6 \)
2. \( y + 5 = 0 \) → \( y = -5 \)

Since \( y \) represents a square root, it must be non-negative. Therefore, we discard \( y = -5 \).

### Step 6: Conclusion
The value of \( y \) is:
\[ y = 6 \]

Thus, the value of \( \sqrt{30 + \sqrt{30 + \sqrt{30 + \ldots}}} \) is \( 6 \).

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