What is the value of 666+ + +⋅⋅⋅

Question

What is the value of 666+ + +⋅⋅⋅

1. A. 3 (Correct answer)
2. B. 2
3. C. 5
4. D. 6

Correct Answer

Option A — 3

Detailed Solution & Explanation

The correct answer is 3.

Key Points

• This is an infinite nested radical: y = √(6 + √(6 + √(6 + …))).
• Because the pattern repeats, the expression inside equals y itself, so y = √(6 + y).
• Squaring: y² = 6 + y → y² − y − 6 = 0 → (y − 3)(y + 2) = 0. Since a square root is non-negative, reject y = −2, leaving y = 3.
• For any √(a + √(a + …)) the shortcut is to solve y² = a + y and keep the positive root.