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)求数列an和bn的通项公式; (2)设数列bn/an的前n项和Tn,问是否存在正整数m、M且M-m=3,使得m
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![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)求数列an和bn的通项公式; (2)设数列bn/an的前n项和Tn,问是否存在正整数m、M且M-m=3,使得m](/uploads/image/z/3274988-68-8.jpg?t=%E5%B7%B2%E7%9F%A5%E6%95%B0%E5%88%97an%2Cbn%E6%BB%A1%E8%B6%B3a1%3D2%2F3%2Can%2B1%3D2an%2Fan%2B2%2Cb1%2B2b2%2B2%5E2b3%2B%2B2%5En-1bn%3Dn%28nN%2A%29+%281%29%E6%B1%82%E6%95%B0%E5%88%97an%E5%92%8Cbn%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%9B+%282%29%E8%AE%BE%E6%95%B0%E5%88%97bn%2Fan%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8CTn%2C%E9%97%AE%E6%98%AF%E5%90%A6%E5%AD%98%E5%9C%A8%E6%AD%A3%E6%95%B4%E6%95%B0m%E3%80%81M%E4%B8%94M-m%3D3%2C%E4%BD%BF%E5%BE%97m)
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)求数列an和bn的通项公式; (2)设数列bn/an的前n项和Tn,问是否存在正整数m、M且M-m=3,使得m
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)求数列an和bn的通项公式; (2)设数列bn/an的前n项和Tn,问是否存在正整数m、M且M-m=3,使得m
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)求数列an和bn的通项公式; (2)设数列bn/an的前n项和Tn,问是否存在正整数m、M且M-m=3,使得m
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