已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8,(1)求数列{an}通项公式(2)数列{1/an}的前n项和为Sn,求数列{nSn}的前n项和Tn
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![已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8,(1)求数列{an}通项公式(2)数列{1/an}的前n项和为Sn,求数列{nSn}的前n项和Tn](/uploads/image/z/1097297-17-7.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Blog2%5Ean%7D%EF%BC%88n%E6%98%AF%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2Ca1%3D2%2Ca3%3D8%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Blog2%5Ean%7D%EF%BC%88n%E6%98%AF%E6%AD%A3%E6%95%B4%E6%95%B0%EF%BC%89%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2Ca1%3D2%2Ca3%3D8%2C%EF%BC%881%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%EF%BC%89%E6%95%B0%E5%88%97%7B1%2Fan%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E6%B1%82%E6%95%B0%E5%88%97%7BnSn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CTn)
已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8,(1)求数列{an}通项公式(2)数列{1/an}的前n项和为Sn,求数列{nSn}的前n项和Tn
已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8
已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8,(1)求数列{an}通项公式(2)数列{1/an}的前n项和为Sn,求数列{nSn}的前n项和Tn
已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8已知数列{log2^an}(n是正整数)是等差数列,a1=2,a3=8,(1)求数列{an}通项公式(2)数列{1/an}的前n项和为Sn,求数列{nSn}的前n项和Tn
(1)
设数列{log2(an)}公差为d
log2(a3)-log2(a1)=2d=log2(8)-log2(2)=3-1=2
d=1
log2(an)=log2(a1) +(n-1)d=log2(2)+(n-1)=1+n-1=n
an=2ⁿ
n=1时,a1=2;n=3时,a3=2³=8,同样满足
数列{an}的通项公式为an=2ⁿ
(2)
1/an=1/2ⁿ
Sn=1/a1+1/a2+...+1/an
=1/2 +1/2²+...+1/2ⁿ
=(1/2)(1-1/2ⁿ)/(1-1/2)
=1- 1/2ⁿ
nSn=n -n/2ⁿ
Tn=S1+2S2+...+nSn
=(1+2+...+n) -(1/2 +2/2²+...+n/2ⁿ)
令Cn=1/2 +2/2²+...+n/2ⁿ
则Cn/2=1/2²+2/2³+...+(n-1)/2ⁿ+ n/2^(n+1)
Cn -Cn/2=Cn/2=1/2 +1/2²+...+1/2ⁿ -n/2^(n+1)
=(1/2)(1-1/2ⁿ)/(1-1/2) -n/2^(n+1)
=1- 1/2ⁿ -n/2^(n+1)
Cn=2 -1/2^(n-1) -n/2ⁿ
Tn=(1+2+...+n)-Cn=n(n+1)/2 +n/2ⁿ +1/2^(n-1) -2