数学红对勾例2:若数列{an}满足a1=1,且an+1=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,n∈N﹡,求数列{bn/an}前n项和.网上题目相同,但问的问题不同,请不要直接粘贴.
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![数学红对勾例2:若数列{an}满足a1=1,且an+1=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,n∈N﹡,求数列{bn/an}前n项和.网上题目相同,但问的问题不同,请不要直接粘贴.](/uploads/image/z/15246422-62-2.jpg?t=%E6%95%B0%E5%AD%A6%E7%BA%A2%E5%AF%B9%E5%8B%BE%E4%BE%8B2%EF%BC%9A%E8%8B%A5%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D1%2C%E4%B8%94an%2B1%3Dan%2F%281%2Ban%29%2C%E8%AE%BE%E6%95%B0%E5%88%97%7Bbn%7D%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C%E4%B8%BASn%2C%E4%B8%94Sn%3D2-bn%2Cn%E2%88%88N%EF%B9%A1%2C%E6%B1%82%E6%95%B0%E5%88%97%7Bbn%2Fan%7D%E5%89%8Dn%E9%A1%B9%E5%92%8C.%E7%BD%91%E4%B8%8A%E9%A2%98%E7%9B%AE%E7%9B%B8%E5%90%8C%2C%E4%BD%86%E9%97%AE%E7%9A%84%E9%97%AE%E9%A2%98%E4%B8%8D%E5%90%8C%2C%E8%AF%B7%E4%B8%8D%E8%A6%81%E7%9B%B4%E6%8E%A5%E7%B2%98%E8%B4%B4.)
数学红对勾例2:若数列{an}满足a1=1,且an+1=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,n∈N﹡,求数列{bn/an}前n项和.网上题目相同,但问的问题不同,请不要直接粘贴.
数学红对勾例2:若数列{an}满足a1=1,且an+1=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,n∈N﹡,
求数列{bn/an}前n项和.
网上题目相同,但问的问题不同,请不要直接粘贴.
数学红对勾例2:若数列{an}满足a1=1,且an+1=an/(1+an),设数列{bn}的前n项和为Sn,且Sn=2-bn,n∈N﹡,求数列{bn/an}前n项和.网上题目相同,但问的问题不同,请不要直接粘贴.
a(n+1)=an/(1+an)
1/a(n+1) = (1+an)/an
= 1+1/an
1/a(n+1) -1/an =1
1/an -1/a1=n-1
an = 1/n
Sn=2-bn
n=1, b1= 1
bn = Sn -S(n-1)
=-bn+b(n-1)
bn = (1/2)b(n-1)
= (1/2)^(n-1) .b1
=(1/2)^(n-1)
let
S=1.(1/2)^0+2.(1/2)^1+...+n.(1/2)^(n-1) (1)
(1/2)S= 1.(1/2)^1+2.(1/2)^2+...+n.(1/2)^n (2)
(1)-(2)
(1/2)S= (1+1/2+...+1/2^(n-1)) -n.(1/2)^n
= 2(1-(1/2)^n)-n.(1/2)^n
S = 4(1-(1/2)^n)-2n.(1/2)^n
=4-(2n+4)(1/2)^n
cn =bn/an
=n.(1/2)^(n-1)
c1+c2+...+cn = S =4-(2n+4)(1/2)^n