已知数列an,bn满足a1=2/3,an+1=2an/an+2,b1+2b2+2^2b3++2^n-1bn=n(nN*)

1.
a(n+1)=2an/(an+2)
1/a(n+1)=(an+2)/(2an)=1/an +1/2
1/a(n+1)-1/an=1/2,为定值
1/a1=1/(2/3)=3/2,数列{1/an}是以3/2为首项,1/2为公差的等差数列
1/an=3/2+(1/2)(n-1)=(n+2)/2
an=2/(n+2)
n=1时,a1=2/(1+2)=2/3,同样满足通项公式
数列{an}的通项公式为an=2/(n+2)
n=1时,b1=1
n≥2时,
b1+2b2+...+2^(n-1)bn=n (1)
b1+2b2+...+2^(n-2)b(n-1)=n-1 (2)
(1)-(2)
2^(n-1)bn=1
bn=1/2^(n-1)
n=1时,b1=1/1=1,同样满足通项公式
数列{bn}的通项公式为bn=1/2^(n-1)
2.
bn/an=[1/2^(n-1)]/[2/(n+2)]=(n+2)/2ⁿ
Tn=b1/a1+b2/a2+...+bn/an
=3/2+4/2^2+5/2^3+...+(n+2)/2ⁿ
Tn/2=3/2^2+4/2^3+...+(n+1)/2ⁿ+(n+2)/2^(n+1)
Tn-Tn/2=Tn/2=3/2+1/2^2+1/2^3+...+1/2ⁿ -(n+2)/2^(n+1)
Tn=3+1/2+1/2^2+...+1/2^(n-1) -(n+2)/2ⁿ
=1+1/2+...+1/2^(n-1) -(n+2)/2ⁿ +2
=1×(1-1/2ⁿ)/(1-1/2) -(n+2)/2ⁿ +2
=4- (n+4)/2ⁿ
(n+4)/2ⁿ>0 4-(n+4)/2ⁿ