Question

What is the value of $\sqrt{50 + \sqrt{50 + \sqrt{50 + \ldots}}}$? 
 

• A. 5
• B. 6
• C. 7
• D. 8

Hint:

For infinite nested radicals, set the whole expression equal to $x$ and solve the resulting equation.

Answer 1

The Correct Option is C

- 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).