{an}为等差数列,an不等于0,d为公差,求证:1/(a1a2)+1/(a2a3)+...+1/(an-1*an)=(n-1)/(aian)

题目

{an}为等差数列,an不等于0,d为公差,求证:1/(a1a2)+1/(a2a3)+...+1/(an-1*an)=(n-1)/(aian)

答案

证明：左边＝1/(a1a2)+1/(a2a3)+...+1/(an-1*an)＝1/d(1/a1-1/a2)+1/d(1/a2-1/a3)+...+1/d(1/an-1-1/an)=1/d[(1/a2-1/a1)+(1/a3-1/a2)+...+(1/an-1-1/an)]=1/d[1/a1+(1/a2-1/a2)+(1/a3-1/a3)+...+(1/an-1 - 1/an-1)-...