Mary's radical

When we normally see any kind of radical problem our instincts are to isolate and raise the radical to its respective power. In this case we can square both sides to obtain $x+\sqrt{x+\sqrt{x+\cdots}}=289$ . Substituting in $\sqrt{x+\sqrt{x+\cdots}}=17$ we find that the answer is $272$

Given that:

$\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = 17$

It can be written that:

$\sqrt{x + 17} = 17$

Squaring both sides, we get that:

$x + 17 = 289 \Rightarrow x = 289 - 17 = \fbox{272}$

Let $y=\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}$ . We can write this as $y=\sqrt{x+y}$ . Squaring both sides , we get $y^2=x+y$ . Rearranging the terms , we get the quadratic in $y$ as $y^2-y-x=0$ . Applying the quadratic formula , we get $y=\frac{1\pm \sqrt{1+4x}}{2}$ . We are given the value of the nested radical as 17 ,i.e. $y=17$ .Therefore , $\frac{1\pm \sqrt{1+4x}}{2}=17$

$\Rightarrow 1\pm \sqrt{1+4x}=34 \Rightarrow \pm\sqrt{1+4x}=33$

Again squaring both sides , we get $1+4x=1089$

$\Rightarrow 4x=1088 \Rightarrow x=\boxed{272}$

From the original equation $$\sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = 17,$$ we square both sides to get $$x + \sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = 289.$$ Subtracting both sides by $x$ we have $$\sqrt{x + \sqrt{x + \sqrt{x + \dotsb}}} = 289 - x.$$ The left side of the equation is equal to $17$ from the first equation so that $289 - x = 17$ from which we get $x = 272$ .

As this is infinitely nested, all $\sqrt{x + \sqrt{x + ...}}$ are equal to 17, so we will substitute the value at the top level.

$n = 17$

$\sqrt{x + n} = n$

$x + n = n^2$

$x + 17 = 289$

$x = 272$

First square the equation: $x+\sqrt{x+\sqrt{x+...}}=289$ . We notice that the infinite $\sqrt{x+\sqrt{x+...}}=17$ , so our new equation becomes

$x+17=289$ .

Then, this implies that $\boxed{x=272}$

To begin with, note that any infinitely nested radical of the type $\sqrt{r+\sqrt{r+\sqrt{r+\dots}}}=\frac{1}{2}(1+\sqrt{1+4n})$ .

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Thus, we have

$\frac{1}{2}(1+\sqrt{1+4x})=17 \Rightarrow \sqrt{1+4x}=33 \Rightarrow 1+4x=1089 \Rightarrow$

$\Rightarrow 4x=1088 \Rightarrow x=272$

Therefore, for the given equation to be equal to $17$ , $x$ needs to be $272$ .

√(x+√(x+√(x+...)))=17 square both side

x+17=289

x=272