(1)
Sn=2n^2+n
S(n-1)=2(n-1)^2+n-1
=2n^2-4n+2+n-1
=2n^2-3n+1
Sn-S(n-1)
=an
=2n^2+n-2n^2+3n-1
=4n-1
an=4log2 (bn)+3=4n-1
4log2(bn)=4n-4
log2(bn)=n-1
bn=2^(n-1)
(2)
an*bn=(4n-1)2^(n-1)=4n*2^(n-1)-2^(n-1)
=n2^(n-1+2)-2^(n-1)
=n2^(n+1)-2^(n-1)
Tn=1*2^2-2^0+2*2^3-2^1+.+n2^(n+1)-2^(n-1)
=1*2^2+2*2^3+.+n2^(n+1)-(2^0+2^1+.+2^(n-1))
∵2^0+2^1+.+2^(n-1)是首項(xiàng)k1=1,公比q=2的等比數(shù)列
2^0+2^1+.+2^(n-1)
=1*(2^n-1)/(2-1)
=2^n-1
設(shè)
Mn=1*2^2+2*2^3+.+n2^(n+1)
2Mn=2*2^2+2*2^4+.+n2^(n+2)
兩式相減得
2Mn-Mn=Mn
=1*2^3+2*2^4+.+n2^(n+2)-1*2^2-2*2^3-.-n2^(n+1)
=-2^2-2^3-.-2^(n+1)+n2^(n+2)
∵數(shù)列-2^2-2^3-.-2^(n+1)是首項(xiàng)=-4,公比q=2的等比數(shù)列
-2^2-2^3-.-2^(n+1)
=-4(2^n-1)(2-1)
=-4(2^n-1)
所以
Mn=n2^(n+2)-4(2^n-1)
=n2^(n+2)-2^(n+2)+4
=(n-1)2^(n+2)+4