Question

Rút gọn biểu thức:

\(\sqrt{x+\sqrt{x+...+\sqrt{x+\frac{1+\sqrt{4x+1}}{2}}}}\), (n dấu căn)

Answer 1

\(x+\frac{1+\sqrt{4x+1}}{2}=\frac{2x+1+\sqrt{4x+1}}{2}=\frac{\left(4x+1\right)+2\sqrt{4x+1}+1}{4}=\left(\frac{1+\sqrt{4x+1}}{2}\right)^2\)

=> \(\sqrt{x+\frac{1+\sqrt{4x+1}}{2}}=\sqrt{\left(\frac{1+\sqrt{4x+1}}{2}\right)^2}=\frac{1+\sqrt{4x+1}}{2}\). tiếp tục n dấu căn

=> A = \(\frac{1+\sqrt{4x+1}}{2}\)