数列{an}中,a1=3,Nan+1-(n+1)an=2n·(n+1)1求证{an/n}为等差数列,冰球通项公式an.2设bn=(an-2n^2)·3^n,求数列{bn}的前n项和S.
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![数列{an}中,a1=3,Nan+1-(n+1)an=2n·(n+1)1求证{an/n}为等差数列,冰球通项公式an.2设bn=(an-2n^2)·3^n,求数列{bn}的前n项和S.](/uploads/image/z/10465481-65-1.jpg?t=%E6%95%B0%E5%88%97%7Ban%7D%E4%B8%AD%2Ca1%3D3%2CNan%2B1-%EF%BC%88n%2B1%EF%BC%89an%3D2n%C2%B7%EF%BC%88n%2B1%EF%BC%891%E6%B1%82%E8%AF%81%7Ban%2Fn%7D%E4%B8%BA%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E5%86%B0%E7%90%83%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8Fan.2%E8%AE%BEbn%3D%EF%BC%88an-2n%5E2%EF%BC%89%C2%B73%5En%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CS.)
数列{an}中,a1=3,Nan+1-(n+1)an=2n·(n+1)1求证{an/n}为等差数列,冰球通项公式an.2设bn=(an-2n^2)·3^n,求数列{bn}的前n项和S.
数列{an}中,a1=3,Nan+1-(n+1)an=2n·(n+1)
1求证{an/n}为等差数列,冰球通项公式an.
2设bn=(an-2n^2)·3^n,求数列{bn}的前n项和S.
数列{an}中,a1=3,Nan+1-(n+1)an=2n·(n+1)1求证{an/n}为等差数列,冰球通项公式an.2设bn=(an-2n^2)·3^n,求数列{bn}的前n项和S.
1.
nan+1-(n+1)an=2n·(n+1)
等号两边同除n(n+1)得an+1/(n+1)-an/n=2
所以{an/n}为等差数列,公差为2
an/n=a1/1+2(n-1)=3+2(n-1)=2n+1,得出an=n(2n+1)=2n^2+n
2.
bn=(an-2n^2)·3^n=(2n^2+n-2n^2)3^n=n·3^n
所以数列{bn}的前n项和S=1*3^1+2*3^2+……+n·3^n
等式两边同乘以3得3S=1*3^2+2*3^3+……+n·3^(n+1)
两式相减得-2S=3^1+3^2+……+3^n-n·3^(n+1)
=3(1-3^n)/(1-3)-n·3^(n+1)
=1/2*3^(n+1)-3/2-n·3^(n+1)
=(1/2-n)3^(n+1)-3/2
所以S=1/4(2n-1)3^(n+1)+3/4