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,

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