已知数列{an}的前n项和为Sn,且a1=1/2,a(n+1)=(n+1)an/2n,(1)求{an}的通项公式;(2)设bn=n(2-Sn),n属于N*,若集合M={n|bn大于等于μ,n属于N*}恰有4个元素,求实数μ的取值范围.

Question

已知数列{an}的前n项和为Sn,且a1=1/2,a(n+1)=(n+1)an/2n,(1)求{an}的通项公式;(2)设bn=n(2-Sn),n属于N*,若集合M={n|bn大于等于μ,n属于N*}恰有4个元素,求实数μ的取值范围.

Answer 1

(1)
a(n+1)=(n+1)an/(2n)
a(n+1)/(n+1) = (1/2) (an/n)
{an/n} 是等比数列,q=1/2
an/n = (1/2)^(n-1) .( a1/1)
= (1/2)^n
an = n.(1/2)^n
(2)
let
S = 1.(1/2)^1+2(1/2)^2+.+n.(1/2)^n (1)
(1/2)S = 1.(1/2)^2+2(1/2)^3+.+n.(1/2)^(n+1) (2)
(1) -(2)
(1/2)S = (1/2 + 1/2^2+...+1/2^n)-n(1/2)^(n+1)
= (1-1/2^n) - n(1/2)^(n+1)
S = 2 - (n+2)(1/2)^n
Sn =a1+a2+...+an
= S
= 2 - (n+2)(1/2)^n
bn = n(2-Sn)
= n(n+2)(1/2)^n
let
f(x) = x(x+2) (1/2)^x
f'(x) =( -x(x+2)ln2 + (2x+2) ) (1/2)^x =0
-x(x+2)ln2 + (2x+2)=0
(ln2)x^2 -(2-2ln2)x - 2 =0
x = 1.31
b1= 3(1/2)^1 = 3/2
b2 = 8(1/2)^2 = 2
max bn= b2 = 2
b3 = 15(1/8) = 15/8
b4 = 24(1/16) = 3/2
b5 = 35/32
M={n|bn>μ,n属于N*}恰有4个元素
35/32