S2=a1+a2=a2+1=4a1+1=4+1=5
a2=4
Sn=4a(n-1)+1
S(n+1)=4an+1
S(n+1)-Sn=a(n+1)=4an-4a(n-1)
a(n+1)-2an=2an-4a(n-1)
[a(n+1)-2an]/[an-2a(n-1)]=2,为定值.
a2-2a1=4-2=2
数列{a(n+1)-2an}是以2为首项,2为公比的等比数列.
bn=a(n+1)-2an
数列{bn}是以2为首项,2为公比的等比数列.
a(n+1)-2an=2ⁿ
等式两边同除以2ⁿ
a(n+1)/2ⁿ -an/2^(n-1)=1,为定值.
a1/2^0=1/1=1
数列{an/2^(n-1)}是以1为首项,1为公差的等差数列.
an/2^(n-1)=1+(n-1)=n
an=n×2^(n-1)
n=1时,a1=1×1=1,同样满足
数列{an}的通项公式为an=n×2^(n-1)