MHB Can You Tackle This Infinite Nested Radical Equation?

[anemone]

Here is this week's POTW:

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Solve the equation $\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{2^{4016}x+3}}}}}=\sqrt{x}+1$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[anemone]

Congratulations to the following members for their correct solution::)

1. MarkFL
2. greg1313
3. lfdahl
4. Sudharaka
5. kaliprasad

Solution from Sudharaka:

Spoiler

Squaring the equation yields,

\[\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{2^{4016}x+3}}}}=2\sqrt{x}+1=\sqrt{4x}+1\]

Squaring again yields,

\[\sqrt{16x+\sqrt{\cdots+\sqrt{2^{4016}x+3}}}=4\sqrt{x}+1=\sqrt{16x}+1\]

Continuing this process we will get,

\[\sqrt{2^{4016}x+3}=\sqrt{2^{4016}x}+1\]

\[\Rightarrow 2^{4016}x+3=2^{4016}x+2\sqrt{2^{4016}x}+1\]

\[\Rightarrow x=\frac{1}{2^{4016}}\]