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.