已知数列an满足a1=0且Sn+1=2Sn+1/2n(n+1) 1,求 a2 a3,并证明an+1=2an+n 2,设b

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  • S(n+1)=2Sn+(1/2)n(n+1)

    1,求 a2 a3,并证明a(n+1)=2an+n

    S2=2S1+1=1

    a2=s2-a1=1

    S3=2S2+3=2+3=5

    a3=s3-s2=5-1=4

    所以a2=1,a3=4;

    S(n+1)=2Sn+(1/2)n(n+1)

    Sn=2S(n-1)+(1/2)n(n-1)

    两式相减:

    a(n+1)=2an+(1/2)n(n+1)-(1/2)n(n-1)

    =2an+n.

    2,设bn=a(n+1)-an,求证b(n+1)=2bn+

    a(n+1)=2an+n

    a(n+1)+(n+1)=2an+2n+1

    [a(n+1)+(n+1)]=2(an+n)+1

    [a(n+1)+(n+1)+1]=2(an+n+1)

    令cn=an+n+1

    c(n+1)=2cn

    cn=2^(n-1)c1=2^(n-1) (a1+1+1)=2^n

    an+n+1=2^n

    an=-n-1+2^n

    bn=a(n+1)-an

    =(2an+n)-an

    =an+n

    =(-n-1+2^n)+n

    =-1+2^n

    b(n+1)=-1+2^(n+1)

    =-1+2*2^n

    =-1+2*(bn+1)

    =2bn +1

    第2题简单方法:

    bn=a(n+1)-an

    =(2an+n)-an

    =an+n

    an=bn-n

    a(n+1)=b(n+1)-(n+1)

    b(n+1)-(n+1)=a(n+1)

    =2an+n

    =2(bn-n)+n

    =2bn-n

    b(n+1)-(n+1)=2bn-n

    b(n+1)=2bn+1