(1)
Sn=-n²/2 +kn
=(-1/2)(n²-2kn+k²)+k²/2
=(-1/2)(n-k)²+k²/2
当n=k时,Sn有最大值(Sn)max=k²/2=8
k²=16
k=-4(k为自然数,舍去)或k=4
k=4
(2)
Sn=-n²/2 +4n
n=1时,a1=S1=-1/2 +4=7/2
n≥2时,
Sn=-n²/2 +4n S(n-1)=-(n-1)²/2 +4(n-1)
an=Sn-S(n-1)=-n²/2 +4n +(n-1)²/2 -4(n-1)
= -n +9/2
=(9-2n)/2
n=1时,a1=(9-2)/2=7/2,同样满足.
数列{an}的通项公式为an=(9-2n)/2
(9-2an)/2ⁿ=[9-2(9-2n)/2]/2ⁿ
=n/2^(n-1)
Tn=1/2^0 +2/2^1+3/2^2+...+n/2^(n-1)
Tn/2=1/2^1+2/2^2+...+(n-1)/2^(n-1)+n/2ⁿ
Tn-Tn/2=Tn/2=1/2^0+1/2^1+1/2^2+...+1/2^(n-1) -n/2ⁿ
=[1-(1/2)ⁿ]/(1-1/2) -n/2ⁿ
=2-2/2ⁿ-n/2ⁿ
=2-(n+2)/2ⁿ
Tn=4- (n+2)/2^(n-1)