数列an满足a1=1,a2=1/3,并且an(an-1+an+1)=2an+1an-1.则数列第2012项为?

题目

数列an满足a1=1,a2=1/3,并且an(an-1+an+1)=2an+1an-1.则数列第2012项为?

答案

an(an-1+an+1)=2an+1an-1 an*an-1+an+1*an=an+1*an-1+an+1*an-1
（ an-an+1）*an-1=（an-1-an）*an+1 an/an+1 -1=1- an/an-1
都除以an得1/an+1 - 1/an = 1/an - 1/an-1
所以｛1/an｝是等差数列 1/a1 =1,d=1/a2 - 1/a1=3-1=2
1/an = 1/a1 +(n-1)d
1/a2012 = 1+(2012-1) *2=4023 a2012 = 1/4023