“已知数列an的前n项和为sn,且满足Sn=2an-1,数列bn满足b1=l,nbn+丨=（n+1）bn.求bn,an

Question

“已知数列an的前n项和为sn,且满足Sn=2an-1,数列bn满足b1=l,nbn+丨=（n+1）bn.求bn,an

Answer 1

n=1时,a1=S1=2a1-1
a1=1
n≥2时,
an=Sn-S(n-1)=2an-1-2a(n-1)+1
an=2a(n-1)
an/a(n-1)=2,为定值
数列{an}是以1为首项,2为公比的等比数列
an=1×2^(n-1)=2^(n-1)
nb(n+1)=(n+1)bn
b(n+1)/(n+1)=bn/n
b1/1=1/1=1,数列{bn/n}是各项均为1的常数数列
bn/n=1
bn=n
数列{an}的通项公式为an=2^(n-1)；数列{bn}的通项公式为bn=n