问题标题：
各项均为正数的数列{an}的前n项和为Sn满足2Sn=an(an+1),n∈N*,求an,我想问(2)设cn={a·n为奇数;3*2^(a)+1·n为偶数},求前2n项的和T
问题描述：
各项均为正数的数列{an}的前n项和为Sn满足2Sn=an(an+1),n∈N*,求an
,我想问(2)设cn={a·n为奇数;3*2^(a)+1·n为偶数},求前2n项的和T
黄龙杨回答：
(1)
2Sn=an(an+1)(1)
n=1
2a1=a1(a1+1)
a1^2-a1=0
a1=1
2S(n-1)=a(n-1)(a(n-1)+1)(2)
(1)-(2)
2an=(an)^2+an-[a(n-1)]^2-a(n-1)
(an)^2-[a(n-1)]^2-[an+a(n-1)]=0
[an+a(n-1)].[an-a(n-1)-1]=0
an-a(n-1)-1=0
an-a(n-1)=1
{an}是等差数列,d=1
an-a1=n-1
an=n
(2)
cn=a(n+1);nisodd
=3.2^(a(n-1))+1;niseven
c1+c3+...+c(2n-1)
=a2+a4+...+a(2n)
=2+4+...+2n
=(n+1)n(3)
c2+c4+.+c(2n)
=(3.2^1+1)+(3.2^3+1)+.+(3.2^(2n-1)+1)
=n+2[2^(2n)-1](4)
T(2n)=c1+c2+.+c(2n)
=(n+1)n+n+2[2^(2n)-1]
=(n+2)n+2[2^(2n)-1]
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