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Question Number 195435 by universe last updated on 02/Aug/23

$$\sqrt{5+1\sqrt{6+2\sqrt{7+3\sqrt{8+4\sqrt{...}}}}}=?$$

Answered by mr W last updated on 02/Aug/23

$$3=\sqrt{9}=\sqrt{5+4}$$$$4=1\times 4=1\times \sqrt{16}=1\times \sqrt{6+10}$$$$10=2\times 5=2\times \sqrt{25}=2\times \sqrt{7+18}$$$$18=3\times 6=3\times \sqrt{36}=3\times \sqrt{8+28}$$$$28=4\times 7=4\times \sqrt{49}=4\times \sqrt{9+40}$$$$40=5\times 8=5\times \sqrt{64}=5\times \sqrt{10+54}$$$$54=6\times 9=6\times \sqrt{81}=6\times \sqrt{11+70}$$$$...$$$$\Rightarrow 3=\sqrt{5+1\sqrt{6+2\sqrt{7+3\sqrt{8+4\sqrt{9+5\sqrt{10+...}}}}}}$$

Commented by Frix last updated on 02/Aug/23

You ��️ are too fast for me ��

Answered by Frix last updated on 02/Aug/23

$$3$$$$\sqrt{5+1\sqrt{6+2\sqrt{7+3\sqrt{8+4\sqrt{...}}}}}=3$$$$1\sqrt{6+2\sqrt{7+3\sqrt{8+4\sqrt{...}}}}=4$$$$2\sqrt{7+3\sqrt{8+4\sqrt{...}}}=2\times 5$$$$3\sqrt{8+4\sqrt{...}}=3\times 6$$$$4\sqrt{...}=4\times 7$$$$...$$

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