已知数列{an}、{bn}满足a1=b1=6,a2=b2=4,a3=b3=3,且{an+1-an}(n∈Z)是等差数列,{bn-2}(n∈Z)是等比数列. (1)求数列{bn}的通项公式; (2)求数列{an}的通项公式; (3)是否存在k

问题描述:

已知数列{an}、{bn}满足a1=b1=6,a2=b2=4,a3=b3=3,且{an+1-an}(n∈Z)是等差数列,{bn-2}(n∈Z)是等比数列.
(1)求数列{bn}的通项公式;
(2)求数列{an}的通项公式;
(3)是否存在k∈Z+,使ak-bk(0,

1
2
)?若存在,求出k的值;若不存在,说明理由.

(1)∵{bn-2} (n∈Z+)为等比数列,又b1-2=4,b2-2=2,b3-2=1,∴公比q=12,bn−2=4•(12)n−1,bn=2+4•(12)n−1(n∈Z+)(2分)(2)∵{an+1-an} (n∈Z+)是等差数列,又a2-a1=-2,a3-a2=-1,∴...