Multiple Choice Question (MCQ)
If x=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\infty}}}} , then the value of x is
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×
\dfrac{\sqrt{7}+1 }{2}
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×
\dfrac{\sqrt{6}+1 }{2}
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✓
\dfrac{\sqrt{5}+1 }{2}
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×
\dfrac{\sqrt{5}-1 }{2}
Answer
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If x=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\infty}}}} , the radical part under the first radical is also equal to x. Then we get the following equation.
x=\sqrt{1+x}
x^2-x-1=0
Solving the quadratic equation gives
x = \dfrac{\sqrt{5}+1 }{2}
Therefore C is the answer