What is the value of $\sqrt{50 + \sqrt{50 + \sqrt{50 + \ldots}}}$?
What is the value of $\sqrt{50 + \sqrt{50 + \sqrt{50 + \ldots}}}$?
Show Hint
- 5
- 6
- 7
8
The Correct Option is C
Solution and Explanation
- Step 1: Let the expression be $x$ - \[ x = \sqrt{50 + \sqrt{50 + \sqrt{50 + \ldots}}} \]
- Step 2: Recognize the infinite nature - The nested radical repeats itself, so: \[ x = \sqrt{50 + x} \]
- Step 3: Square both sides - \[ x^2 = 50 + x \]
- Step 4: Rearrange - \[ x^2 - x - 50 = 0 \]
- Step 5: Solve quadratic - Discriminant $\Delta = 1 + 200 = 201$: \[ x = \frac{1 \pm \sqrt{201}}{2} \] Since $x$ is positive, take $x = \frac{1 + \sqrt{201}}{2} \approx 7.58$. Closest integer is 7.
- Step 6: Conclusion - Value is approximately 7, matching option (3).
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