Question:easy

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

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Let the whole expression equal x, then square both sides to form a quadratic equation.
Updated On: Jul 21, 2026
  • \( -7 \)
  • \( -6 \)
  • \( 6 \)
  • \( 7 \)
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The Correct Option is D

Solution and Explanation

Step 1: Build the sequence of partial values.
Let $a_1 = \sqrt{42}$, $a_2 = \sqrt{42 + a_1}$, $a_3 = \sqrt{42 + a_2}$, and so on, each term nesting one more root.

Step 2: Estimate the first few terms.
$a_1 = \sqrt{42} \approx 6.48$, $a_2 = \sqrt{42 + 6.48} \approx 6.96$, $a_3 = \sqrt{42 + 6.96} \approx 6.997$, clearly closing in on 7.

Step 3: Confirm the limit algebraically.
If the sequence converges to a value $L$, then $L = \sqrt{42 + L}$, so $L^2 - L - 42 = 0$, which factors as $(L-7)(L+6)=0$.
Since $L$ is a square root, it cannot be negative, so $L = 7$.

Final Answer:
The nested radical converges to 7. \[ \boxed{7} \]
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