已知数列{an},{bn}的前n项和分别为An,Bn,且A100=8,B100=251.记Cn=an•Bn+bn•An-

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  • 解题思路:由题意可知Cn=An×Bn-An-1×Bn-1,Cn=C1+C2+…+Cn-1+Cn=a1×b1+(A2×B2-a1×b1)+…+(An-1×Bn-1-An-2×Bn-2)+(An×Bn-An-1×Bn-1)=An×Bn,由此可以求出数列{Cn}的前100项的和.

    Cn=an•Bn+bn•An-an•bn

    =(An-An-1)×Bn+(Bn-Bn-1)×An-(An-An-1)×(Bn-Bn-1

    =An×Bn-An-1×Bn+Bn×An-Bn-1×An-(An×Bn-An-1×Bn-An×Bn-1+An-1×Bn-1]

    =An×Bn-An-1×Bn-1

    ∴Cn=An×Bn-An-1×Bn-1

    Cn-1=An-1×Bn-1-An-2×Bn-2

    …C2=A2×B2-a1×b1

    C1=a1×b1

    ∴Cn=C1+C2+…+Cn-1+Cn

    =a1×b1+(A2×B2-a1×b1)+…+(An-1×Bn-1-An-2×Bn-2)+(An×Bn-An-1×Bn-1)=An×Bn

    ∴C100=A100×B100=8×251=2008

    C(100)=A(100)×B(100)=8×251=2008.

    答案:2008.

    点评:

    本题考点: 数列的应用.

    考点点评: 本题考查数列的性质和应用,解题时要注意培养学生的计算能力.