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