The correct answer is option (D):
7
Let x = √(42 + √(42 + √(42 + √(42 + ...)))). Notice that the expression inside the outermost square root is the same as the entire expression itself.
We can rewrite the equation as x = √(42 + x).
To solve for x, square both sides of the equation:
x² = 42 + x
Rearrange the equation into a quadratic equation:
x² - x - 42 = 0
Factor the quadratic equation:
(x - 7)(x + 6) = 0
This gives us two possible solutions for x: x = 7 and x = -6. Since the original expression involves the square root of a number, and square roots always yield non-negative values, the value of the nested radical expression must also be non-negative. Therefore, we discard the solution x = -6.
The only valid solution is x = 7. Thus, the value of the given expression is 7.