Question

已知数列an满足1/a1a2+1/a2a3+……1/an-1an=(n-1)/a1an,求证为等差数列

Answer 1

1/(a1a2)+1/(a2a3)+……+1/(a<n-1>an)=(n-1)/(a1an),①
以n+1代n,得1/(a1a2)+1/(a2a3)+……+1/(a<n-1>an)+1/(ana<n+1>)=n/(a1a<n+1>),②
②-①，1/(ana<n+1>)=n/(a1a<n+1>)-(n-1)/(a1an)，
去分母得a1=nan-(n-1)a<n+1>,
∴an-a1=(n-1)(a<n+1>-an),
∴a<n+1>-an=（an-a1)/(n-1),设a2-a1=d,则
a3-a2=（a2-a1)/(2-1)=d,
a4-a3=(a3-a1)/(3-1)=(a3-a2+a2-a1)/2=(d+d)/2=d,
依此类推，a<n+1>-an=d,
∴{an}是等差数列。