n=1時(shí),a1=S1=2^1 -1=2-1=1
n≥2時(shí),
Sn=2^n -1
Sn-1=2^(n-1) -1
Sn-Sn-1=an=2^n -1-2^(n-1) +1=2^(n-1)
n=1時(shí),a1=2^(1-1)=2^0=1,同樣滿足.
數(shù)列{an}的通項(xiàng)公式為an=2^(n-1)
b(n+1)-2bn=8×2^(n-1)=2^(n+2)
b(n+1)=2bn+2^(n+2)
b(n+1)/2^(n+1)=2bn/2^(n+1)+2^(n+2)/2^(n+1)
b(n+1)/2^(n+1)=bn/2^n +2
[b(n+1)/2^(n+1)]-(bn/2^n)=2,為定值.
b1/2^1=2/2=1
數(shù)列{bn/2^n}是以1為首項(xiàng),2為公差的等差數(shù)列.
bn/2^n=1+2(n-1)=2n-1
bn=(2n-1)×2^n
n=1時(shí),b1=(2-1)×2^1=2,同樣滿足.
數(shù)列{bn}的通項(xiàng)公式為bn=(2n-1)×2^n.
Tn=b1+b2+...+bn=1×2^1+3×2^2+5×2^3+...+(2n-1)×2^n
2Tn=1×2^2+3×2^3+...+(2n-3)×2^n+(2n-1)×2^(n+1)
Tn-2Tn=-Tn=2^1+2×2^2+2×2^3+...+2×2^n-(2n-1)×2^(n+1)
=2+2×4×[2^(n-1) -1]/(2-1) -(2n-1)×2^(n+1)
=(3-2n)×2^(n+1)-6
Tn=(2n-3)×2^(n+1) +6.
^表示指數(shù).