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/

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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)

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)+···+1/(根号a(n+1)+根号an)
=( (根号a2 - 根号a1) + (根号a3 - 根号a2) + ...+ (根号a(n+1) - 根号an )/d
= (根号a(n+1) - 根号a1)/d
= n(根号a(n+1) - 根号a1) / (nd)
= n(根号a(n+1)-根号a1) / (a(n+1) - a1)
= n / (根号a(n+1) + 根号a1)

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