已知数列{an}的各项均为正数,其前n项和为Sn.且满足2Sn=an^2+an(n∈N*).求数列an的通项公式若bn=n(1/2)^an,求数列{bn}的前n项和Tn.非常急,求能人十分钟内解答
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![已知数列{an}的各项均为正数,其前n项和为Sn.且满足2Sn=an^2+an(n∈N*).求数列an的通项公式若bn=n(1/2)^an,求数列{bn}的前n项和Tn.非常急,求能人十分钟内解答](/uploads/image/z/1270228-4-8.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97%7Ban%7D%E7%9A%84%E5%90%84%E9%A1%B9%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%2C%E5%85%B6%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn.%E4%B8%94%E6%BB%A1%E8%B6%B32Sn%3Dan%5E2%2Ban%28n%E2%88%88N%2A%29.%E6%B1%82%E6%95%B0%E5%88%97an%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E8%8B%A5bn%3Dn%EF%BC%881%2F2%EF%BC%89%5Ean%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CTn.%E9%9D%9E%E5%B8%B8%E6%80%A5%2C%E6%B1%82%E8%83%BD%E4%BA%BA%E5%8D%81%E5%88%86%E9%92%9F%E5%86%85%E8%A7%A3%E7%AD%94)
已知数列{an}的各项均为正数,其前n项和为Sn.且满足2Sn=an^2+an(n∈N*).求数列an的通项公式若bn=n(1/2)^an,求数列{bn}的前n项和Tn.非常急,求能人十分钟内解答
已知数列{an}的各项均为正数,其前n项和为Sn.且满足2Sn=an^2+an(n∈N*).求数列an的通项公式
若bn=n(1/2)^an,求数列{bn}的前n项和Tn.
非常急,求能人十分钟内解答
已知数列{an}的各项均为正数,其前n项和为Sn.且满足2Sn=an^2+an(n∈N*).求数列an的通项公式若bn=n(1/2)^an,求数列{bn}的前n项和Tn.非常急,求能人十分钟内解答
n=1时,2S1=2a1=a1²+a1
a1²-a1=0 a1(a1-1)=0
a1=0(各项均为正数,舍去)或a1=1
n≥2时,
2Sn=an²+an
2Sn-1=a(n-1)²+a(n-1)
2Sn-2Sn-1=2an=an²+an-a(n-1)²-a(n-1)
an²-a(n-1)²-an-a(n-1)=0
[an+a(n-1)][an-a(n-1)]-[an+a(n-1)]=0
[an+a(n-1)][an-a(n-1)-1]=0
数列各项均为正,an+a(n-1)恒>0,要等式成立,只有an-a(n-1)=1,为定值.
数列{an}是以1为首项,1为公差的等差数列.
an=1+n-1=n
数列{an}的通项公式为an=n
bn=n×(1/2)^an=n/2^n
Tn=b1+b2+b3+...+bn=1/2^1+2/2^2+3/2^3+...+n/2^n
Tn/2=1/2^2+2/2^3+...+(n-1)/2^n+n/2^(n+1)
Tn-Tn/2=Tn/2=1/2^1+1/2^2+1/2^3+...+1/2^n -n/2^(n+1)
=(1/2)(1-1/2^n)/(1-1/2) -n/2^(n+1)
=1-1/2^n -n/2^(n+1)
Tn=2 -1/2^(n-1) -n/2^n