Question

设数列{an}的前n项和为bn,数列{bn}的前n项和为cn,且bn+cn=n

Answer 1

(1)

bn+cn=n

b(n+1)+c(n+1)=n+1

相减 得 b(n+1)-bn+b(n+1)=1

1-bn=2-2b(n+1)

[1-b(n+1)]/1-bn=1/2

公比1/2

bn+cn=n 代入n=1

b1=1/2

1-b1=1/2

所以{1-bn}是首项1/2,公比1/2的等比数列

(2)

1-bn=(1/2)^n

bn=1-(1/2)^n

cn=b1+b2+…bn=n-[1/2+(1/2)^2+…+(1/2)^n]

=n-[1-(1/2)^n]

=n-1+(1/2)^n

Sn=1+2…+n-n+1/2+(1/2)^2+…+(1/2)^n

=n(n-1)/2+[1-(1/2)^n]

=n(n-1)/2-(1/2)^n+1