Nested radical: disprove

Algebra Level 4

(Identify the incorrect step.)

Is

x+x+x+=1+4x+12\sqrt{x + \sqrt{x + \sqrt{x + \ldots }}} = \frac{-1 + \sqrt{4x + 1}}{2}?

Here is the proof ( with x > 0)

Let

y=x+x+x+\displaystyle y = \sqrt{x + \sqrt{x + \sqrt{x + \ldots }}}


Step 1:

Multiply "i" on both the side( i=1 i = \sqrt{-1})

yi=ix+x+x+\displaystyle yi = i\sqrt{x + \sqrt{x + \sqrt{x + \ldots }}}

Step 2:

Take "i" inside the root

yi=xi2+i2x+x+\displaystyle yi = \sqrt{xi^{2} + i^{2}\sqrt{x + \sqrt{x + \ldots }}}

Step 3:

Take i2i^{2} inside the root

yi=xi2+xi4+i4x+\displaystyle yi = \sqrt{xi^{2} + \sqrt{xi^{4} + i^{4}\sqrt{x + \ldots }}}

Step 4:

Squaring both the sides

y2=x+x+x+\displaystyle -y^{2} = -x + \sqrt{ x + \sqrt{x + \ldots}}

Step 5:

Solving the above quadratic

y2=x+y\displaystyle -y^{2} = -x + y

y2+yx=0\displaystyle y^{2} + y - x = 0

y=1+4x+12\boxed{ y = \frac{-1 + \sqrt{4x + 1}}{2}}( neglecting -ve sign as x > 0)


(Again, identify the incorrect step.)

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