已知数列{an}的前项和为Sn,a1=1/4,且2Sn=2S(n-1)+2a(n-1)+1,(n≧2)数列{bn}满足b
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已知数列{an}的前项和为Sn,a1=1/4,且2Sn=2S(n-1)+2a(n-1)+1,(n≧2)数列{bn}满足b1=3/4,且3bn-b(n-1)=n,(n≧2) ①求证数列{bn-an}为等比数列,②求数列{Bn}的通项公式以及前n项和Tn
3bn-3(1/2)n+3/4=b(n-1)-(n-1)/2+1/4怎么来的,
3bn-3(1/2)n+3/4=b(n-1)-(n-1)/2+1/4怎么来的,
![已知数列{an}的前项和为Sn,a1=1/4,且2Sn=2S(n-1)+2a(n-1)+1,(n≧2)数列{bn}满足b](/uploads/image/z/4558621-13-1.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8D%E9%A1%B9%E5%92%8C%E4%B8%BASn%2Ca1%EF%BC%9D1%2F4%2C%E4%B8%942Sn%EF%BC%9D2S%28n%EF%BC%8D1%29%EF%BC%8B2a%28n%EF%BC%8D1%29%EF%BC%8B1%2C%28n%E2%89%A72%29%E6%95%B0%E5%88%97%7Bbn%7D%E6%BB%A1%E8%B6%B3b)
2Sn=2Sn-1+2a(n-1)+1
2Sn-2Sn-1=2an=2a(n-1)+1
an-a(n-1)=1/2,为定值.
a1=1/4
数列{an}是以1/4为首项,1/2为公差的等差数列
an=1/4+(n-1)(1/2)=n/2 -1/4
3bn-b(n-1)=n
3bn-3(1/2)n+3/4=b(n-1)-(n-1)/2+1/4
[bn-n/2 +1/4]/[b(n-1)-(n-1)/2+1/4]=1/3,为定值.
b1-1/2+1/4=3/4-1/2+1/4=1/2
数列{bn-n/2 +1/4}是以1/2为首项,1/3为公比的等比数列
bn- n/2+1/4=(1/2)(1/3)^(n-1)
bn=n/2 +(1/2)(1/3)^(n-1) -1/4
bn-an=n/2+(1/2)(1/3)^(n-1) -1/4 -n/2+1/4=(1/2)(1/3)^(n-1)
b1-a1=3/4-1/4=1/2,同样满足.
(bn-an)/[b(n-1)-a(n-1)]=1/3,为定值.
数列{bn-an}是以1/2为首项,1/3为公比的等比数列.
bn=n/2+(1/2)(1/3)^(n-1) -1/4
Tn=(1+2+...+n)/2+(1/2)[1-(1/3)^n]/(1-1/3)-n/4
=n^2/4-(1/4)/3^(n-1)+3/4
2Sn-2Sn-1=2an=2a(n-1)+1
an-a(n-1)=1/2,为定值.
a1=1/4
数列{an}是以1/4为首项,1/2为公差的等差数列
an=1/4+(n-1)(1/2)=n/2 -1/4
3bn-b(n-1)=n
3bn-3(1/2)n+3/4=b(n-1)-(n-1)/2+1/4
[bn-n/2 +1/4]/[b(n-1)-(n-1)/2+1/4]=1/3,为定值.
b1-1/2+1/4=3/4-1/2+1/4=1/2
数列{bn-n/2 +1/4}是以1/2为首项,1/3为公比的等比数列
bn- n/2+1/4=(1/2)(1/3)^(n-1)
bn=n/2 +(1/2)(1/3)^(n-1) -1/4
bn-an=n/2+(1/2)(1/3)^(n-1) -1/4 -n/2+1/4=(1/2)(1/3)^(n-1)
b1-a1=3/4-1/4=1/2,同样满足.
(bn-an)/[b(n-1)-a(n-1)]=1/3,为定值.
数列{bn-an}是以1/2为首项,1/3为公比的等比数列.
bn=n/2+(1/2)(1/3)^(n-1) -1/4
Tn=(1+2+...+n)/2+(1/2)[1-(1/3)^n]/(1-1/3)-n/4
=n^2/4-(1/4)/3^(n-1)+3/4
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