Question

已知数列an满足a1=4,an=4 - 4/an-1 (n>1),记bn= 1 / an-2 .(1)求证:数列bn是等差数列

Answer 1

证明：因为bn中有an-2出现，所以要使an出现an-2形式，所以：
原式
=an-2
= 4-4/a(n-1)-2
= 2-4/a(n-1)
= [2a(n-1)-4]/a(n-1)1/(an-2)
= a(n-1)/[2a(n-1)-4]
= [a(n-1)-2+2])/2[a(n-1)-2]
= 1/2-1/[a(n-1)-2]1/(an-2)-1/[a(n-1)-2]
= 1/2bn-b(n-1)
= 1/2为常数，所以bn为等差数列