数列{an}的前n项和为Sn=1-2/3an(nN+)证明数列an是等比数列.球数列an的前n项和Sn

题目

数列{an}的前n项和为Sn=1-2/3an(nN+)证明数列an是等比数列.球数列an的前n项和Sn

答案

S1=a1=1-(2/3)a1
(5/3)a1=1
a1=3/5
Sn=1-(2/3)an
Sn-1=1-(2/3)a(n-1)
Sn-Sn-1=an=1-(2/3)an-1+(2/3)a(n-1)
5an=3a(n-1)
an/a(n-1)=3/5
数列{an}是以3/5为首项,3/5为公比的等比数列.
an=(3/5)^n
Sn=(3/5)[1-(3/5)^n]/(1-3/5)=3/2 -(3/2)(3/5)^n=3/2 - 3^(n+1)/(2×5^n)