设正数a1,a2,a3,···an成等差数列,求证:1/(根号a1+根号a2)+1/(根号a2+根号a3)+···+1/(根号an+a(n-1)
题目
设正数a1,a2,a3,···an成等差数列,求证:1/(根号a1+根号a2)+1/(根号a2+根号a3)+···+1/(根号an+a(n-1)
==n/(根号a1+a(n+1)
答案
证明,假设等差数列的公差为d.因为1/(根号a1 + 根号a2)= (根号a2 - 根号a1) / (a2-a1)= (根号a2 - 根号a1) / d同理可得1/(根号a2 + 根号a3)= (根号a3 - 根号a2) / d所以类似的有1/(根号a1+根号a2)+1/(根号a2+根号a3)+...
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