Question

Cho :   \(A=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}\)

CM   \(A=2\)

Please help me!!!!

Answer 1

\(taco:A^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+....}}}}\Rightarrow A^2-A=2\Rightarrow A=2\left(dpcm\right)\)

Answer 2

\(A=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdot\cdot\cdot}}}}\)

\(\Rightarrow A^2=2+\sqrt{2+\sqrt{2+\sqrt{2+\cdot\cdot\cdot}}}\)

\(\Rightarrow A^2-A=\left(2+\sqrt{2+\sqrt{2+\cdot\cdot\cdot}}\right)-\left(\sqrt{2+\sqrt{2+\cdot\cdot\cdot\cdot}}\right)\)

\(\Rightarrow A=2\)

Answer 3

\(A^2=2+\sqrt{2+\cdot\cdot\cdot}\)

\(\Rightarrow A^2-A=\left(2+\sqrt{2+\cdot\cdot\cdot}\right)-\left(\sqrt{2+\sqrt{2+\cdot\cdot\cdot}}\right)\)

\(\Rightarrow A=2\)