已知各项均为正数的数列{an}的前项和为Sn,且Sn,an,1/2成等差数列.(1)求a1,a2的值;(2)求数列{an}的通项公式;(3) 若bn=4-2n(n∈N+),设Cn=bn/an,求数列{cn}的前n项和Tn
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![已知各项均为正数的数列{an}的前项和为Sn,且Sn,an,1/2成等差数列.(1)求a1,a2的值;(2)求数列{an}的通项公式;(3) 若bn=4-2n(n∈N+),设Cn=bn/an,求数列{cn}的前n项和Tn](/uploads/image/z/8883042-42-2.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%84%E9%A1%B9%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%E7%9A%84%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%89%8D%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%94Sn%2Can%2C1%2F2%E6%88%90%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97.%281%29%E6%B1%82a1%2Ca2%E7%9A%84%E5%80%BC%EF%BC%9B%282%29%E6%B1%82%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B%283%29+%E8%8B%A5bn%3D4-2n%EF%BC%88n%E2%88%88N%2B%EF%BC%89%2C%E8%AE%BECn%3Dbn%2Fan%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bcn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CTn)
已知各项均为正数的数列{an}的前项和为Sn,且Sn,an,1/2成等差数列.(1)求a1,a2的值;(2)求数列{an}的通项公式;(3) 若bn=4-2n(n∈N+),设Cn=bn/an,求数列{cn}的前n项和Tn
已知各项均为正数的数列{an}的前项和为Sn,且Sn,an,1/2成等差数列.
(1)求a1,a2的值;
(2)求数列{an}的通项公式;
(3) 若bn=4-2n(n∈N+),设Cn=bn/an,求数列{cn}的前n项和Tn
已知各项均为正数的数列{an}的前项和为Sn,且Sn,an,1/2成等差数列.(1)求a1,a2的值;(2)求数列{an}的通项公式;(3) 若bn=4-2n(n∈N+),设Cn=bn/an,求数列{cn}的前n项和Tn
1
Sn+1/2=2an
S1=a1 a1+1/2=2a1,a1=1/2
(a2+1/2)+1/2=2a2
a2=1
2
Sn-1+1/2=2an-1
an=Sn-Sn-1=2(an-an-1)
an/an-1=2
an=a1*2^(n-1)
=(1/2)*2^(n-1)=2^(n-2)
3
cn=4/2^(n-2)-2n/2^(n-2)
=16/2^n-n/2^(n-3)
c1=16/2-1/2^(-2)
Tn=(16/2+16/4+...+16/2^n) - (1/2^(-2)+2/2^(-1)+...+n/2^(n-3))
设tn'=(1/2^(-2)+2/2^(-1)+...+n/2^(n-3))
2tn'=(1/2^(-1)+2+3/2^1+..+n/2^(n-2))
(tn'-n/2^(n-3)) -2tn' =1/2^(-2)+1/2^(-1)+1+1/2+..+1/2^(n-2)
tn'=(1/4)(1-1/2^n)/(1-1/2)-n/2^(n-3)
=1/2-1/2^(n+1)-n/2^(n-3)
Tn=16*(1/2)(1-2^n)/(1-1/2) + 1/2-1/2^(n+1)-n/2^(n-3)
=16-2^(n+4)+1/2-1/2^(n+1)-n/2^(n-3)