已知数列An前n项的和Sn,且Sn+1/2An=1,求An的通项

Question

已知数列An前n项的和Sn,且Sn+1/2An=1,求An的通项

Answer 1

【分析】此题用到的数列公式是：An=Sn - S 和 S1=A1
另外,本题题目有点不清楚,是表示Sn+(1/2)*An=1,还是表示Sn + 1/(2*An)=1.我姑且按第一种理解Sn+(1/2)*An=1来算,如果我理解错了,你再追问（以后抄题必要处记得加个括号）~
∵Sn + (1/2)*An=1………………………………………………………………①
∴S + (1/2)*A=1…………………………………………………………②
根据公式,An=Sn - S
则① - ②,得
An + (1/2)*An - (1/2)*A=0
化简得,An = (1/3)*A
又由①得,S1 + (1/2)A1 =1,而S1=A1
代入解得,A1 = 2/3
∴An = (1/3)*A = (1/3)²*A
=……= 【(1/3)^(n-1)】*A1
= 【(1/3)^(n-1)】*(2/3)
= 2 ÷ ( 3^n )