一道高二数学解答题已知函数f(x)=log以a为底x的对数(a>0且≠1),若数列2,f(a1),f(a2),...,f

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  • (1)f(an)=2n+2 即 loga(an)=2n+2

    an=a^(2n+2)(2n+2次方)

    Sn=a1+a2+……+an=a^4+a^6+……+a^(2n+2)

    a^2Sn=a^6+a^8+……+a^(2n+2)+a^(2n+4)

    两式相减得(a^2-1)Sn=a^(2n+4)-a^4=a^(2n+4)-a^4

    Sn=[a^(2n+4)-a^4]/(a^2-1)

    2)bn=an*f(an)=a^(2n+2)*loga[a^(2n+2)]= (2n+2)*a^(2n+2)

    设vn=2*a^2+4*a^4+……+2n*a^2n

    Tn =b1+b2+……+bn=3a^3+6a^6+……+(2n+2)*a^(2n+2)

    =a^2*vn+2Sn 故只要求出vn即可

    a^2*vn=2*a^4+4*a^6+……+2n*a^(2n+2)

    两式相减得(1-a^2)vn=2*a^2+2*a^4+2*a^6+……+2*a^2n-2n*a^(2n+2)=[2a^(2n+2)-2a^2]/(a^2-1)-2n*a^(2n+2)/(a^2-1)

    Tn =a^2*vn+2Sn =4*[a^(2n+4)-a^4]/(a^2-1))-2n*a^(2n+2)/(a^2-1)