Ҳисоб кунед:

\(\sqrt{2+\sqrt{2+\sqrt{2+...}}}\)

\(x=\sqrt{2+\sqrt{2+\sqrt{2+...}}}\)

\(x^2=2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}\)

\(x^2-x=2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}-\sqrt{2+\sqrt{2+\sqrt{2+...}}}\)

\(x^2-x=2\)

\(x^2-x-2=0\)

\(D=1+8=9\)

\(x_1=\frac{1+\sqrt{9}}{2}=\frac{4}{2}=2\)

\(x_2=\frac{1-\sqrt{9}}{2}=\frac{-2}{2}=-1\)

Азбаски x>0, x=2

\(\sqrt{2+\sqrt{2+\sqrt{2+...}}}=2\)

Ҷавоб: 2.