已知三个正数a,b,c成等差数列,求证:1/(根号b+根号c),1/(根号c+根号a),1/(根号a+根号b)成等差数列

设a-b=b-c=d
1/(sqrt(b)+sqrt(c))+1/(sqrt(a)+sqrt(b))
=(sqrt(b)-sqrt(c))/(b-c)+(sqrt(a)-sqrt(b))/(a-c)
=(sqrt(b)-sqrt(c)+sqrt(a)-sqrt(b))/d
=2*(sqrt(a)-sqrt(c))/(2*d)
=2*(sqrt(a)-sqrt(c))/(a-c)
=2/(sqrt(a)+sqrt(c))
因此
1/(sqrt(b)+sqrt(c))-1/(sqrt(a)+sqrt(c))=1/(sqrt(a)+sqrt(c))-1/(sqrt(a)+sqrt(b))
证毕