已知数列an的通项公式an=(4n-5)*(1/2)^(n-1),试猜测an的最大值并通过研究数列an的单调性证明结论,

来源：学生作业帮编辑：大师作文网作业帮分类：数学作业时间：2024/09/25 23:25:36

已知数列an的通项公式an=(4n-5)*(1/2)^(n-1),试猜测an的最大值并通过研究数列an的单调性证明结论,并求Sn

f(x) = (4x-5).(1/2)^(x-1)
f'(x) = (1/2)^(x-1) .[ 4 - (4x-5) ln2] =0
4 - (4x-5) ln2=0
x= (4+5ln2)/(4ln2) =2.69
an=(4n-5)*(1/2)^(n-1)
a2= 3(1/2) = 3/2
a3 = 7(1/2)^2 = 7/4
max an = a3 = 7/4
an is increasing 1≤n≤3
an is decreasing n ≥3
an = (4n-5).(1/2)^(n-1)
= 8(n.(1/2)^n) - 5(1/2)^(n-1)
Sn = 8[∑(i:1->n) i.(1/2)^i] - 10[ 1-(1/2)^n]
let
S = 1.(1/2)+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)^3+...+(1/2)^n] - n.(1/2)^(n+1)
= 1-(1/2)^n - n.(1/2)^(n+1)
S = 2[1-(1/2)^n - n.(1/2)^(n+1)]
Sn = 8[∑(i:1->n) i.(1/2)^i] - 10[ 1-(1/2)^n]
=8S -10[ 1-(1/2)^n]
=16[1-(1/2)^n - n.(1/2)^(n+1)] - 10[ 1-(1/2)^n]
= 6 -2(4n+3).(1/2)^n

已知数列an的通项公式an=(4n-5)*(1/2)^(n-1),试猜测an的最大值并通过研究数列an的单调性证明结论, 已知an=n×0.8^n判断并证明数列的单调性,求数列的最大值已知an=n-根号1+n^2,判断数列（an)的单调性数列an的前n项和为Sn=2^n-1,设bn满足bn=an+1/an,判断并证明bn 的单调性已知an=n×0.8^n判断并证明数列的单调性,求数列的最大值怎么求单调性已知数列{an}的前n项和sn满足log2(sn+1)=n（1)求通项公式an（2）求an的单调性已知数列{an},a1=2,an+1=an+2n,则数列的通项公式an=? 已知数列{an}满足a1=1,an=(an-1)/3an-1+1,(n>=2,n属于N*),求数列{an}的通项公式已知数列{an}的前n项和Sn=12n-n²,求数列{an}的通项公式,（1）证明数列{an}是等差数列. 证明数列an=(n+2)/(2n²+1）的单调性已知an=n*（0.8^n）,判断并证明数列{an}的单调性,求数列{an}中的最大项数列an中,a1=3,an+an-1+2n-1=0(1)证明：数列an是等比数列,并求an通项公式；（2）求数列an的前