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Question-65753

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Question Number 65753 by Masumsiddiqui399@gmail.com last updated on 03/Aug/19
Answered by mr W last updated on 04/Aug/19
1 ^(∗)) (√(1+1)) (√((√(1+1))+1)) (√((√((√(1+1))+1))+1)) (√((√((√((√(1+1))+1))+1))+1)) ...... (√((√((√((√((√((√(1+1))+1))+1))+1))+1))+1....))=x x^2 −1=x ⇒x^2 −x−1=0 ⇒x=((1+(√5))/2)=ϕ=golden ratio≈1.618 ^(∗)) actually you must not start with “1”, you can start with any number, the final result is always the same.
$$\mathrm{1}\:\:\:^{\left.\ast\right)} \\ $$$$\sqrt{\mathrm{1}+\mathrm{1}} \\ $$$$\sqrt{\sqrt{\mathrm{1}+\mathrm{1}}+\mathrm{1}} \\ $$$$\sqrt{\sqrt{\sqrt{\mathrm{1}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}} \\ $$$$\sqrt{\sqrt{\sqrt{\sqrt{\mathrm{1}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}} \\ $$$$…… \\ $$$$\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\mathrm{1}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}}+\mathrm{1}….}={x} \\ $$$$ \\ $$$${x}^{\mathrm{2}} −\mathrm{1}={x} \\ $$$$\Rightarrow{x}^{\mathrm{2}} −{x}−\mathrm{1}=\mathrm{0} \\ $$$$\Rightarrow{x}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}=\varphi={golden}\:{ratio}\approx\mathrm{1}.\mathrm{618} \\ $$$$ \\ $$$$\:^{\left.\ast\right)} \:{actually}\:{you}\:{must}\:{not}\:{start}\:{with} \\ $$$$“\mathrm{1}'',\:{you}\:{can}\:{start}\:{with}\:{any}\:{number}, \\ $$$${the}\:{final}\:{result}\:{is}\:{always}\:{the}\:{same}. \\ $$
Commented by mr W last updated on 03/Aug/19
i tried with my calculator, i got after a few steps following result
$${i}\:{tried}\:{with}\:{my}\:{calculator},\:{i}\:{got} \\ $$$${after}\:{a}\:{few}\:{steps}\:{following}\:{result} \\ $$
Commented by mr W last updated on 03/Aug/19

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