已知数列{an}满足a1=1/4,a2=3/4,a(n+1)=2an-a(n-1)(n>等于2,n属于N*),数列{bn}满足：b1等于2,n属于N*),数列{bn}的前n项和为Sn(1)求证:数列{an}为等差数列(2)求证:数列{bn-an}为等比数列(3)若当且仅当n=4时,Sn取得

Question

已知数列{an}满足a1=1/4,a2=3/4,a(n+1)=2an-a(n-1)(n>等于2,n属于N*),数列{bn}满足：b1等于2,n属于N*),数列{bn}的前n项和为Sn
(1)求证:数列{an}为等差数列
(2)求证:数列{bn-an}为等比数列
(3)若当且仅当n=4时,Sn取得最小值,求b1的取值范围

Answer 1

1.
a(n+1)=2an-a(n-1)
a(n+1)-an=an-a(n-1)
所以{an}为以1/4为首项,1/2为公差的等差数列
an=n/2-1/4
(2)
bn-an=bn-n/2+1/4,bn=1/3b(n-1)+n/3
b(n+1)-a(n+1)=bn/3+n/3+1/3-n/2-1/2+1/4=bn/3-n/6+1/12=(bn-an)/3
所以,数列{bn-an}为等比数列
(3).
b4=(b1+49)/270
解得:
-184