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

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}}}} \dots \dots}

= \dots \dots \dots \dots = \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + 5 \sqrt{1 + 6 \sqrt{1 + \dots . .}}}}}}

\sqrt{1 + \sqrt{1 + 2 \sqrt{1 + 3 \dots . .}}} = \sqrt{1 + 3} = \sqrt{4} = 2

9 = 1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \dots .}}}

∴ \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{1 + \dots . .}}} = 4

s o

\sqrt{1 + \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{\dots .}}}} + \frac{1}{\sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + 4 \sqrt{\dots . .}}}} + \frac{1}{\sqrt{1 + 3 \sqrt{1 + 4 \sqrt{\dots .}}}}}

= 2 + \frac{1}{3 + \frac{1}{4}} = \frac{30}{13}