Question

\(C/m:\sqrt{6+\sqrt{6+\sqrt{6+...}}}=3\)

Answer 1

Đặt \(A=\sqrt{6+\sqrt{6+\sqrt{6+...}}}=3\)

\(A^2=6+\sqrt{6+\sqrt{6+\sqrt{6+...}}}\)

\(A^2=6+A\)

\(A^2-3A+2A-6=0\)

\(A\left(A-3\right)+2\left(A-3\right)=0\)

\(\Rightarrow\left(A-3\right)\left(A+2\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}A-3=0\\A+2=0\end{cases}\Leftrightarrow\orbr{\begin{cases}A=3\left(n\right)\\A=-2\left(l\right)\end{cases}}}\)

Vậy \(\sqrt{6+\sqrt{6+\sqrt{6+...}}}=3\left(đpcm\right)\)