【急,现求!】观察下列各式及证明过程:根号(1/2-1/3)=1/2根号2/3;根号[1/2(1/3-1/4)]=1/3

首先,题目给的有错误.应该是你输入的问题
√(1/2-1/3)=(1/2)√(2/3)；
√[1/2(1/3-1/4)]=(1/3)√(3/8)；
√[1/3(1/4-1/5)]=(1/4)√(4/15).
那么√[1/4(1/5-1/6)]=(1/5)√(5/24)
因为√[1/4(1/5-1/6)]=√[1/(4*5*6)]=√[5/(4*5^2*6)=(1/5)√(5/24)
通式：
√{(1/n)[1/(n+1)-1/(n+2)]}=[1/(n+1)]√[(n+1)/n(n+2)]
验证：
√{(1/n)[1/(n+1)-1/(n+2)]}=√[1/n(n+1)(n+2)]=√[(n+1)/n(n+1)^2(n+2)]=[1/(n+1)]√[(n+1)/n(n+2)]

1

a1 = √(2 + 2/3) = 2 * √(2/3)
a2 = √(3 + 3/8) = 3 * √(3/8)
a3 = √(4 + 4/15) = 4*√(4/15)
a4 = √(5 + 5/24) = 5 * √(5/24)
an = √{(n-1) + (n-1)/[n * (n+2)]} = (n-1)√[(n-1)/(n^2+2n)]