1/a1a2+1/a2a3+……+1/anan+1=n/a1an+1已知数列{an}是各项不为0的无穷数列,求证:{an}为等差数列的充要条件是1/(a1a2)+1/(a2a3)+……+1/(anan+1)=n/(a1an+1)(n属于N*)

Question

1/a1a2+1/a2a3+……+1/anan+1=n/a1an+1
已知数列{an}是各项不为0的无穷数列,求证:{an}为等差数列的充要条件是1/(a1a2)+1/(a2a3)+……+1/(anan+1)=n/(a1an+1)(n属于N*)

Answer 1

充分性：设公差为d,1/(ai*ai+1)=d/d(ai*ai+1)=(ai+1-ai)/d(ai*ai+1)=1/d*ai-1/d*ai+1
故该求和式=1/d*a1-1/d*an+1=n/(a1an+1)
充分性得证
必要性：第n个求和式-第n-1个求和式,可得1/anan+1=n/a1an+1-(n-1)/a1an
去分母,得：a1=n*an-(n-1)an+1=an-(n-1)(an+1-an),即an-a1=(n-1)(an+1-an)
归纳假设当n≤k时为等差数列,公差为d,则当n=k+1时,由上式知ak-a1=(k-1)(ak+1-ak)
即（k-1）d=(k-1)(ak+1-ak),故ak+1-ak=d
必要性得证