设n为正整数根号n+4-根号n+3与根号n+2-根号n+1比大小

设n为正整数根号n+4-根号n+3与根号n+2-根号n+1比大小
设n为正整数根号n+4-根号n+3与根号n+2-根号n+1比大小

设n为正整数根号n+4-根号n+3与根号n+2-根号n+1比大小
根号n+4-根号n+3=1/[根号(N+4)+根号(N+3)]
根号n+2-根号n+1=1/[根号(N+2)+根号(N+1)]
由于:根号(N+4)+根号(N+3)>根号(N+2)+根号(N+1)
所以,根号n+4-根号n+3 < 根号n+2-根号n+1

因为（√n+4)+（√n+3)>（√n+2）+（√n+1）,又（√n+4)-（√n+3)的平方差=（√n+2）-（√n+1）的平方差, 可算得（√n+4)-（√n+3)>（√n+2）-（√n+1）