已知公差不为0的等差数列{an}满足a2=3,a1,a3,a7成等比数列 (1)求an的通项公式

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  • (Ⅰ)由题知 ax0523 =a1a7,设等差数列{an}的公差为d,

    则(a1+2d)2=a1(a1+6d),

    a1d=2d2,∵d≠0

    ∴a1=2d. …(1分)

    又∵a2=3,

    ∴a1+d=3a1=2,d=1…(2分)

    ∴an=n+1. …(3分)

    (Ⅱ)∵bn= anx05an+1 + an+1x05an = n+1x05n+2 + n+2x05n+1 =2+ 1x05n+1 - 1x05n+2 . …(4分)

    ∴Sn=b1+b2+…+bn=(2+ 1x052 - 1x053 )+(2+ 1x053 - 1x054 )+…+(2+ 1x05n+1 - 1x05n+2 )=2n+ nx052(n+2) . …(6分)

    ( III)cn=2n( an+1x05n -λ)=2n( n+2x05n -λ),使数列{cn}是单调递减数列,

    则cn+1-cn=2n( 2(n+3)x05n+1 - n+2x05n -λ)<0对n∈N*都成立 …(7分)

    即 2(n+3)x05n+1 - n+2x05n -λ<0⇒λ>( 2(n+3)x05n+1 - n+2x05n )max…(8分)

    设f(n)= 2(n+3)x05n+1 - n+2x05n ,

    f(n+1)-f(n)= 2(n+4)x05n+2 - n+3x05n+1 - 2(n+3)x05n+1 + n+2x05n

    = 2(n+4)x05n+2 + n+2x05n - 3(n+3)x05n+1

    =2+ 4x05n+2 +1+ 2x05n -3- 6x05n+1

    = 2(2-n)x05n(n+1)(n+2) …(9分)

    ∴f(1)<f(2)=f(3)>f(4)>f(5)>…

    当n=2或n=3时,f(n)max= 4x053 ,

    ∴( 2(n+3)x05n+1 - n+2x05n )max= 4x053

    所以λ> 4x053 .