Question

CM biểu thức sau là 1 số nguyên: B=\(\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+.....}}}}\)

Answer 1

\(B=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+....+\sqrt{6}}}}}\)

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

\(\Rightarrow B^2=6+B\)

\(\Rightarrow B^2-B-6=0\)

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

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

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

\(\Rightarrow\left[{}\begin{matrix}B+3=0\\B-2=0\end{matrix}\right.\)

\(\Rightarrow B=2\left(B\in Z\right)\)

\(\Rightarrow\) Đpcm.