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/(aan)=(n-1)/(a1an),①

以n+1代n,得1/(a1a2)+1/(a2a3)+……+1/(aan)+1/(ana)=n/(a1a),②

②-①,1/(ana)=n/(a1a)-(n-1)/(a1an),

去分母得a1=nan-(n-1)a,

∴an-a1=(n-1)(a-an),

∴a-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-an=d,

∴{an}是等差数列.