Question

The value of \( \sqrt{6 + \sqrt{6 + \sqrt{6 + \cdots}}} \) is:

• A. 3
• B. 2
• C. 6
• D. 1

Hint:

For $\sqrt{n(n+1) + \sqrt{\dots}}$, the answer is always $(n+1)$. Here $6 = 2 \times 3$, so answer is 3.

Answer 1

The Correct Option is A

Step 1: Concept
Let the expression be $x$. Then $x = \sqrt{6 + x}$.
Step 2: Analysis
Squaring both sides: $x^{2} = 6 + x \Rightarrow x^{2} - x - 6 = 0$.
$(x-3)(x+2) = 0$.
Step 3: Conclusion
$x = 3$ or $x = -2$. Since the square root of positive numbers must be positive, $x = 3$.
Final Answer: (A)