Question

What is the value of \( \sqrt{42 + \sqrt{42 + \sqrt{42 + \sqrt{42 + \sqrt{42 + \ldots \infty}}}}} \)?

• A. \( -7 \)
• B. \( -6 \)
• C. \( 6 \)
• D. \( 7 \)

Hint:

Let the whole expression equal x, then square both sides to form a quadratic equation.

Answer 1

The Correct Option is D

Step 1: Name the whole expression.
Let \( x = \sqrt{42 + \sqrt{42 + \sqrt{42 + \ldots \infty}}} \).
Since the pattern under the first root repeats forever, it equals \( x \) itself, so \( x = \sqrt{42 + x} \).

Step 2: Square both sides.
\( x^2 = 42 + x \).

Step 3: Rearrange into a quadratic.
\( x^2 - x - 42 = 0 \).

Step 4: Factor and solve.
\( (x - 7)(x + 6) = 0 \), so \( x = 7 \) or \( x = -6 \).

Step 5: Reject the invalid root.
A square root is never negative, so \( x = -6 \) is rejected, leaving \( x = 7 \).

Final Answer:
The value of the nested radical is 7. \[ \boxed{7} \]