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Question Number 204664 by cortano12 last updated on 25/Feb/24

$$8+\sqrt{8^2+\sqrt{8^4+\sqrt{8^8+\sqrt{8^{16}+\sqrt{...}}}}}=?$$

Answered by Frix last updated on 25/Feb/24

$$=8+8\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}$$$$=8+8\phi =12+4\sqrt{5}$$

Commented by cortano12 last updated on 25/Feb/24

$$\phi =\mathrm{golden}\mathrm{ratio}$$

Answered by SEKRET last updated on 25/Feb/24

$$8+\sqrt{8\cdot 8+8\sqrt{8^2+\sqrt{8^4+\sqrt{8^8+....}}}}=\mathbf{X}$$$$8+\sqrt{8}\cdot \sqrt{8+\sqrt{8^2+\sqrt{8^4+..{.}^{}}}}=\mathbf{X}$$$$8+\sqrt{8}\cdot \sqrt{\mathbf{X}}=\mathbf{X}$$$$\mathbf{X}=12+4\sqrt{5}$$$$$$

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