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Question Number 145918 by mathdanisur last updated on 09/Jul/21

Answered by Olaf_Thorendsen last updated on 09/Jul/21

$$l=3+\sqrt{3^2+\sqrt{3^4+\sqrt{3^8+\sqrt{...}}}}$$$${(l-3)}^2=3^2+\sqrt{3^4+\sqrt{3^8+\sqrt{...}}}$$$${(l-3)}^2=3^2+\sqrt{3^4+3^2\sqrt{3^4+\sqrt{...}}}$$$${(l-3)}^2=3^2+3\sqrt{3^2+\sqrt{3^4+\sqrt{...}}}$$$$\frac{{(l-3)}^2}{3}=3+\sqrt{3^2+\sqrt{3^4+\sqrt{...}}}=l$$$$l^2-9l+9=0$$$$l=\frac{9\pm \sqrt{81-4\times 1\times 9}}{2}$$$$l=\frac{9\pm 3\sqrt{5}}{2}$$$$\frac{9-3\sqrt{5}}{2}\approx 1,146:\mathrm{impossible}\mathrm{because}l>3$$$$\mathrm{Finally}l=\frac{9+3\sqrt{5}}{2}\approx 7,854$$

Commented by Mrsof last updated on 09/Jul/21

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Commented by Snail last updated on 09/Jul/21

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Commented by mathdanisur last updated on 09/Jul/21

$$ThankyouSer,cool$$

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