Question:medium

What is the value of \(\sqrt{42+\sqrt{42+\sqrt{42+\sqrt{42+\sqrt{42+...\infin}}}}}\) ?

Updated On: Jul 23, 2026
  • -7
  • -6
  • 6
  • 7
  • 42
Show Solution

The Correct Option is D

Solution and Explanation

The correct answer is option (D):
7

Let x = sqrt(42 + sqrt(42 + sqrt(42 + sqrt(42 + sqrt(42 + ...))))).

Since the pattern continues infinitely, we can rewrite the expression as:
x = sqrt(42 + x)

To solve for x, we can square both sides of the equation:
x^2 = 42 + x

Now, rearrange the equation into a quadratic equation:
x^2 - x - 42 = 0

We can factor this quadratic equation:
(x - 7)(x + 6) = 0

This gives us two possible solutions for x:
x = 7 or x = -6

However, since the original expression involves a square root, the result must be a non-negative number. Therefore, we discard the negative solution.

Thus, the value of the expression is 7.
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