Question

15．觀察并驗證下列算式：①$\sqrt{1+\frac{1}{3}}$=2$\sqrt{\frac{1}{3}}$，②$\sqrt{2+\frac{1}{4}}$=3$\sqrt{\frac{1}{4}}$，③$\sqrt{3+\frac{1}{5}}$=4$\sqrt{\frac{1}{5}}$…由此規(guī)律猜想第n個算式為：$\sqrt{n+\frac{1}{n+2}}=（n+1）\sqrt{\frac{1}{n+2}}$．

Answer 1

分析 根據(jù)觀察等式：①$\sqrt{1+\frac{1}{3}}$=2$\sqrt{\frac{1}{3}}$，②$\sqrt{2+\frac{1}{4}}$=3$\sqrt{\frac{1}{4}}$，③$\sqrt{3+\frac{1}{5}}$=4$\sqrt{\frac{1}{5}}$，找出規(guī)律解答即可．

解答 解：根據(jù)觀察等式找出規(guī)律可猜想，第n個算式為：
$\sqrt{n+\frac{1}{n+2}}=（n+1）\sqrt{\frac{1}{n+2}}$．
故答案為：$\sqrt{n+\frac{1}{n+2}}=（n+1）\sqrt{\frac{1}{n+2}}$．

點評 本題考查了二次根式的性質(zhì)與化簡，解答本題的關鍵在于認真觀察算式并找出規(guī)律．