设数列{an}的前n项和为Sn=2n^2,{bn}为等比数列,且a1=b1,b2(a2-a1)=b1.(1)求数列{an}和{bn}的通项公式;(2)设Cn=an/bn,求数列{Cn}的前n项和Tn.
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![设数列{an}的前n项和为Sn=2n^2,{bn}为等比数列,且a1=b1,b2(a2-a1)=b1.(1)求数列{an}和{bn}的通项公式;(2)设Cn=an/bn,求数列{Cn}的前n项和Tn.](/uploads/image/z/644167-55-7.jpg?t=%E8%AE%BE%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%3D2n%5E2%2C%EF%BD%9Bbn%EF%BD%9D%E4%B8%BA%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97%2C%E4%B8%94a1%3Db1%2Cb2%28a2-a1%29%3Db1.%281%29%E6%B1%82%E6%95%B0%E5%88%97%EF%BD%9Ban%EF%BD%9D%E5%92%8C%EF%BD%9Bbn%EF%BD%9D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B%EF%BC%882%EF%BC%89%E8%AE%BECn%3Dan%2Fbn%2C%E6%B1%82%E6%95%B0%E5%88%97%EF%BD%9BCn%EF%BD%9D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CTn.)
设数列{an}的前n项和为Sn=2n^2,{bn}为等比数列,且a1=b1,b2(a2-a1)=b1.(1)求数列{an}和{bn}的通项公式;(2)设Cn=an/bn,求数列{Cn}的前n项和Tn.
设数列{an}的前n项和为Sn=2n^2,{bn}为等比数列,且a1=b1,b2(a2-a1)=b1.
(1)求数列{an}和{bn}的通项公式;
(2)设Cn=an/bn,求数列{Cn}的前n项和Tn.
设数列{an}的前n项和为Sn=2n^2,{bn}为等比数列,且a1=b1,b2(a2-a1)=b1.(1)求数列{an}和{bn}的通项公式;(2)设Cn=an/bn,求数列{Cn}的前n项和Tn.
a1=S1=2
Sn=2n^2
Sn-1=2(n-1)^2=2n^2-4n+2
an=Sn-Sn-1=2n^2-2n^2+4n-2=4n-2
n=1代入4-2=2=a1,同样满足.
数列{an}通项公式为an=4n-2
b1=a1=2
a2=4×2-2=6
b2(a2-a1)=b1
b2(6-2)=2
b2=1/2
b2/b1=(1/2)/2=1/4
数列{bn}是以2为首项,1/4为公比的等比数列.
bn=2(1/4)^(n-1)=8/4^n
数列{bn}的通项公式为bn=8/4^n
cn=an/bn=(4n-2)/[8/4^n]=(2n-1)4^n/4=2n4^(n-1)-4^(n-1)
Tn=2[1×4^0+2×4^1+3×4^2+...+n×4^(n-1)]-(4^n-1)/(4-1)
令Mn=1×4^0+2×4^1+3×4^2+...+n×4^(n-1)
则4Mn=4^1+2×4^2+3×4^3+...+(n-1)×4^(n-1)+n×4^n
Mn-4Mn=-3Mn=4^0+4^1+4^2+...+4^(n-1)-n×4^n=(4^n-1)/(4-1)-n4^n
Mn=n4^n/3-(4^n-1)/9
Tn=2n4^n/3-2(4^n-1)/9-(4^n-1)/3
=[6n4^n-2×4^n+2-3×4^n+3]/9
=[(6n-2-3)4^n+5]/9
=[(6n-5)4^n+5]/9