证：
S2=4a1+2=4×1+2=6
n≥2时,
S(n+1)=4an +2=4[Sn -S(n-1)]+2
S(n+1)-2Sn +2=2Sn -4S(n-1)+4
[S(n+1)-2Sn +2]/[Sn -2S(n-1) +2]=2,为定值.
S2-2S1+2=6-2+2=6
数列{S(n+1)-2Sn +2}是以6为首项,2为公比的等比数列.
S(n+1)-2Sn +2=6×2^(n-1)=3×2ⁿ
S(n+1)=2Sn +3×2ⁿ -2
[S(n+1) -2]-(2Sn -4)=3×2ⁿ
等式两边同除以2ⁿ
[S(n+1)-2]/2ⁿ -(Sn -2)/2^(n-1) =3,为定值.
(S1 -2)/2^0=(1-2)/1=-1
数列{(Sn -2)/2^(n-1)}是以-1为首项,3为公差的等差数列.
(Sn -2)/2^(n-1)=(-1)+3(n-1)=3n-4
Sn=(3n-4)×2^(n-1) +2
n≥2时,
an=Sn -S(n-1)
=(3n-4)×2^(n-1) +2-[3(n-1)-4]×2^(n-2) -2
=(3n-1)×2^(n-2)
n=1时,a1=(3-1)/2=1,同样满足.
数列{an}的通项公式为an=(3n-1)×2^(n-2)
b1=a2-2a1=S2-3a1=6-3=3
bn=a(n+1)-2an
=[3(n+1)-1]×2^(n+1-2) -2[(3n-1)×2^(n-2)]
=3×2^(n-1)
b(n+1)/bn=3×2ⁿ/[3×2^(n-1)]=2,为定值.
数列{bn}是以3为首项,2为公比的等比数列.