已知等差数列an的前n项和为Sn,且对于任意的正整数n满足2根号下Sn=(an)+11)求an通项公式(2)设bn=1/ana(n+1),求数列bn的前n项和Bn(1)an=2n+1出错了(1)an=2n-1
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![已知等差数列an的前n项和为Sn,且对于任意的正整数n满足2根号下Sn=(an)+11)求an通项公式(2)设bn=1/ana(n+1),求数列bn的前n项和Bn(1)an=2n+1出错了(1)an=2n-1](/uploads/image/z/1741115-11-5.jpg?t=%E5%B7%B2%E7%9F%A5%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97an%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%94%E5%AF%B9%E4%BA%8E%E4%BB%BB%E6%84%8F%E7%9A%84%E6%AD%A3%E6%95%B4%E6%95%B0n%E6%BB%A1%E8%B6%B32%E6%A0%B9%E5%8F%B7%E4%B8%8BSn%3D%28an%29%2B11%EF%BC%89%E6%B1%82an%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%29%E8%AE%BEbn%3D1%2Fana%28n%2B1%29%2C%E6%B1%82%E6%95%B0%E5%88%97bn%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CBn%EF%BC%881%EF%BC%89an%3D2n%2B1%E5%87%BA%E9%94%99%E4%BA%86%EF%BC%881%EF%BC%89an%3D2n-1)
已知等差数列an的前n项和为Sn,且对于任意的正整数n满足2根号下Sn=(an)+11)求an通项公式(2)设bn=1/ana(n+1),求数列bn的前n项和Bn(1)an=2n+1出错了(1)an=2n-1
已知等差数列an的前n项和为Sn,且对于任意的正整数n满足2根号下Sn=(an)+1
1)求an通项公式
(2)设bn=1/ana(n+1),求数列bn的前n项和Bn
(1)an=2n+1
出错了(1)an=2n-1
已知等差数列an的前n项和为Sn,且对于任意的正整数n满足2根号下Sn=(an)+11)求an通项公式(2)设bn=1/ana(n+1),求数列bn的前n项和Bn(1)an=2n+1出错了(1)an=2n-1
1.
2√Sn=an+1
4Sn=(an)^2+2an+1
4S1=(a1)^2+2a1+1=4a1,
a1=1
4S(n-1)=[a(n-1)]^2+2a(n-1)+1
4an=4[sn-s(n-1)]=(an)^2+2an-[a(n-1)]^2-2a(n-1)
(an)^2-2an-[a(n-1)]^2-2a(n-1)=0
[an+a(n-1)][an-a(n-1)]-2[an+a(n-1)]=0
[an+a(n-1)][an-a(n-1)-2]=0
an+a(n-1)=0或an-a(n-1)-2=0
an=a1(-1)^(n-1)=(-1)^(n-1)
或an=a1+2(n-1)=2n-1
经检验都符合题意
所以
an=(-1)^(n-1)或an=2n-1.
2.
bn=1/[ana(n+1)]
2-1,当an=(-1)^(n-1)时,
bn=1/[ana(n+1)]
=1/{[(-1)^(n-1)][(-1)^n]}
=1/[(-1)^(2n-1)]
=-1
Bn=-1*n=-n;
2-2,当an=2n-1时,
bn=1/[ana(n+1)]
=1/[(2n-1)(2n+1)]
=(1/2)[1/(2n-1)-1/(2n+1)]
2bn=1/(2n-1)-1/(2n+1)
2b(n-1)=1/(2n-3)-1/(2n-1)
2b(n-2)=1/(2n-5)-1/(2n-3)
2b(n-3)=1/(2n-7)-1/(2n-5)
……
2b3=1/5-1/7
2b2=1/3-1/5
2b1=1/1-1/3
两边相加:
2Bn=2[b1+b2+b3+……+b(n-3)+b(n-2)+b(n-1)+bn]
=1-1/(2n+1)
=2n/(2n+1)
Bn=n/(2n+1).
综上所述
an=(-1)^(n-1)时,Bn=-n
an=2n-1时,Bn=n/(2n+1).
bn=1/(2n+1)(2n+3)
=1/2[2/(2n+1)(2n+3)]
=(1/2)[(2n+3)-(2n+1)]/(2n+1)(2n+3)
=(1/2)[1/(2n+1)-1/(2n+3)]
所以Bn=(1/2)[1/3-1/5+1/5-1/7+……+1/(2n+1)-1/(2n+3)]
=(1/2)[1/3-1/(2n+3)]
=n/(6n+9)