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Question Number 204686 by Davidtim last updated on 25/Feb/24
(√(a−(√(a−(√(a−∙∙∙))))))=?
$$\sqrt{{a}−\sqrt{{a}−\sqrt{{a}−\centerdot\centerdot\centerdot}}}=? \\ $$
Commented by A5T last updated on 25/Feb/24
If (√(a−(√(a−(√(a−...)))))) converges, then (√(a−x))=x  ⇒x^2 +x−a=0⇒x=((−1+_− (√(1+4a)))/2)
$${If}\:\sqrt{{a}−\sqrt{{a}−\sqrt{{a}−…}}}\:{converges},\:{then}\:\sqrt{{a}−{x}}={x} \\ $$$$\Rightarrow{x}^{\mathrm{2}} +{x}−{a}=\mathrm{0}\Rightarrow{x}=\frac{−\mathrm{1}\underset{−} {+}\sqrt{\mathrm{1}+\mathrm{4}{a}}}{\mathrm{2}} \\ $$

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