已知无穷数列an,首项a1=3,其前n项和为Sn已知无穷数列{an},首项a1=3,其前n项和为Sn,且a(n+1)=2+(a-1)Sn(a不等于0,1）若{an}的各项和为－8a/3,则a=

Question

已知无穷数列an,首项a1=3,其前n项和为Sn
已知无穷数列{an},首项a1=3,其前n项和为Sn,且a(n+1)=2+(a-1)Sn (a不等于0,1）若{an}的各项和为－8a/3,则a=__

Answer 1

∵a(n+1)=2+(a-1)Sn∴an=2+(a-1)S(n-1)∴a(n+1)-an=(a-1)an∴a(n+1)/an=a∴{an}是以3为首项,公比为a的等比数列∵{an}的各项和为－8a/3∴|a|＜1∴Sn=a1/(1-a)=-8/3a∴3/(1-a)=-8/3a∴a=3/2＞1（舍）a=-3/4∴a=-3/4