Find x:
x = sq.root[1 + sq.root( 1 + sq.root(1 + sq.root(1 +....
"The dots mean that the series go till infinity.
By attempt the answer should be (1 +-sq.root(5))/2 but that is not the answer.
/the answer is $\displaystyle \frac{1 \frac{+}{-} \sqrt(5)}{2} $
The infinite nested radicals, $\displaystyle \sqrt(1 + \sqrt(1 + \sqrt(1 + .... $ = golden ratio.
Since golden ratio says $\displaystyle \frac{a+b}{a} = \frac{a}{b} = x $ then $\displaystyle 1 + \frac{b}{a} = 1 + \frac{1}{x} = x $ so solving $\displaystyle x^2 - x - 1 = 0 $
Hello, Aditya3003!
$\displaystyle \text{Evaluate: }\:x \:=\: \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} $
By attempt the answer should be: $\displaystyle \frac{1\pm\sqrt{5}}{2}$, but that is not the answer.
Since $\displaystyle x$ is obviously positive,
. . you must reject the negative root: $\displaystyle \tfrac{1-\sqrt{5}}{2}$
While solving, you squared the equation, right?
. . This can introduce an extraneous root.