已知数列{an}的前n项和为sn,且a1=1,an+1=二分之一乘sn(n=1.2.3.) (1)求数列{an}等等通项公式(2)当bn=log二分之三(3an+1)时 求证:数列{bn*bn+1分之1}的前n项和,Tn=1+n分之n
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![已知数列{an}的前n项和为sn,且a1=1,an+1=二分之一乘sn(n=1.2.3.) (1)求数列{an}等等通项公式(2)当bn=log二分之三(3an+1)时 求证:数列{bn*bn+1分之1}的前n项和,Tn=1+n分之n](/uploads/image/z/1346024-56-4.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BAsn%2C%E4%B8%94a1%3D1%2Can%2B1%3D%E4%BA%8C%E5%88%86%E4%B9%8B%E4%B8%80%E4%B9%98sn%28n%3D1.2.3.%29+%281%29%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%AD%89%E7%AD%89%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%282%29%E5%BD%93bn%3Dlog%E4%BA%8C%E5%88%86%E4%B9%8B%E4%B8%89%283an%2B1%29%E6%97%B6+%E6%B1%82%E8%AF%81%3A%E6%95%B0%E5%88%97%7Bbn%2Abn%2B1%E5%88%86%E4%B9%8B1%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%2CTn%3D1%2Bn%E5%88%86%E4%B9%8Bn)
已知数列{an}的前n项和为sn,且a1=1,an+1=二分之一乘sn(n=1.2.3.) (1)求数列{an}等等通项公式(2)当bn=log二分之三(3an+1)时 求证:数列{bn*bn+1分之1}的前n项和,Tn=1+n分之n
已知数列{an}的前n项和为sn,且a1=1,an+1=二分之一乘sn(n=1.2.3.) (1)求数列{an}等等通项公式
(2)当bn=log二分之三(3an+1)时 求证:数列{bn*bn+1分之1}的前n项和,Tn=1+n分之n
已知数列{an}的前n项和为sn,且a1=1,an+1=二分之一乘sn(n=1.2.3.) (1)求数列{an}等等通项公式(2)当bn=log二分之三(3an+1)时 求证:数列{bn*bn+1分之1}的前n项和,Tn=1+n分之n
∵a(n+1)=1/2*Sn,a1=1
∴a2=1/2*a1=1/2
a3=1/2*S2=1/2(a1+a2)=3/4
当n≥2时,an=1/2*S(n-1)
∴a(n+1)-an
=1/2*Sn-1/2*S(n-1)
=1/2*[Sn-S(n-1)]=1/2*an
∴a(n+1)=3/2*an
a(n+1)/an=3/2
∵a2=a1=1/2
∴{an}从第2项起为等比数列,公比为3/2
即n≥2时,an=a2*q^(n-2)=1/2*(3/2)^(n-2)
∴数列{an}等等通项公式 为分段形式
an={ 1,(n=1)
{ 1/2*(3/2)^(n-2)
(2)
∵ a(n+1)=1/2*(3/2)^(n-1)
∴3a(n+1)=(3/2)*(3/2)^(n-1)=(3/2)^n
∴bn=log(3/2)[3a(n+1)] =log(3/2)[(3/2)^n)=n
∴1/[(bnb(n+1)]=1/[n(n+1)]=1/n-1/(n+1)
∴Tn=(1-1/2)+(1/2-1/3)+(1/3-1/4)+.+(1/n-1/(n+1))
=1-1/(n+1)
∴Tn=n/(n+1)
a1 = 1,
a2 = 1/2
a3 = 3/4
...
an = 1-1/2^(n-1)