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Question Number 89064 by john santu last updated on 15/Apr/20

$$\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{...}}}}+\frac{1}{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{...}}}}+\frac{1}{\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{...}}}}}}=...$$

Answered by M±th+et£s last updated on 16/Apr/20

$$3=\sqrt{1+8}=\sqrt{1+2\sqrt{16}}=\sqrt{1+2\sqrt{1+3\sqrt{25}}}$$$$=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{36}}}}=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{49}}}}......}$$$$=............=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+.....}}}}}}$$$$\sqrt{1+\sqrt{1+2\sqrt{1+3.....}}}=\sqrt{1+3}=\sqrt{4}=2$$$$9=1+2\sqrt{1+3\sqrt{1+4\sqrt{1+....}}}$$$$\therefore \sqrt{1+3\sqrt{1+4\sqrt{1+.....}}}=4$$$$so$$$$\sqrt{1+\sqrt{1+2\sqrt{1+3\sqrt{....}}}}+\frac{1}{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{.....}}}}+\frac{1}{\sqrt{1+3\sqrt{1+4\sqrt{....}}}}}$$$$=2+\frac{1}{3+\frac{1}{4}}=\frac{30}{13}$$

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