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已知数列{an}中,a1=4,an=3a(n-1)-2(n大于等于2)(1)求证:数列{an-1}是等比数列,并求数列{an}的通项(2)证明:1/a1+1/a2+...+1/an<1/2
题目详情
已知数列{an}中,a1=4,an=3a(n-1)-2(n大于等于2)
(1)求证:数列{an-1}是等比数列,并求数列{an}的通项
(2)证明:1/a1+1/a2+...+1/an<1/2
(1)求证:数列{an-1}是等比数列,并求数列{an}的通项
(2)证明:1/a1+1/a2+...+1/an<1/2
▼优质解答
答案和解析
(1)
an=3a(n-1)-2
an-1 = 3(a(n-1) -1)
(an-1)/(a(n-1) -1)=3
(an-1)/(a1-1) =3^(n-1)
an = 1+ 3^n
(2)
1/an = 1/(1+3^n) < 1/3^n
1/a1+1/a2+..1/an < 1/3 +1/3^2 +...+1/3^n
= (1/3)(1- (1/3)^n )/ (1-1/3)
= (1/2)(1- (1/3)^n )
an=3a(n-1)-2
an-1 = 3(a(n-1) -1)
(an-1)/(a(n-1) -1)=3
(an-1)/(a1-1) =3^(n-1)
an = 1+ 3^n
(2)
1/an = 1/(1+3^n) < 1/3^n
1/a1+1/a2+..1/an < 1/3 +1/3^2 +...+1/3^n
= (1/3)(1- (1/3)^n )/ (1-1/3)
= (1/2)(1- (1/3)^n )
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