Question

通过对上述等式的判断，你发现了什么规律吗？请用含有n得式子将规律表示出来，并说明n的取值范围

√（2+2/3）=2√2/3，√（3+3/8）=3√3/8。请用学过的数学知识说明你所写的式子的正确性

Answer 1

√[n+n/(n^2-1)]=n√[n/(n^2-1)]
证明左边=√【（n^3-n+n）/(n^2-1)】=√[n^2/(n^2-1)]=右边

Answer 2

√[n+n/(n^2-1)]=n*√[n/(n^2-1)] (n>1，n∈N)

证明：左=√[(n^3-n+n)/(n^2-1)]
=√[n^3/(n^2-1)]
=n*√[n/(n^2-1)]
=右。