数列(an)满足a1=1,且8a(n+1)an -16a(n+1) +2an+5=0 ,记bn=1/an-0.5(n大于等于1)求数列(anbn)的通项公式及前n项和Sn (注:an为a的第n项,a(n+1)为a的第n+1项)
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![数列(an)满足a1=1,且8a(n+1)an -16a(n+1) +2an+5=0 ,记bn=1/an-0.5(n大于等于1)求数列(anbn)的通项公式及前n项和Sn (注:an为a的第n项,a(n+1)为a的第n+1项)](/uploads/image/z/7378420-4-0.jpg?t=%E6%95%B0%E5%88%97%EF%BC%88an%EF%BC%89%E6%BB%A1%E8%B6%B3a1%3D1%2C%E4%B8%948a%28n%2B1%29an+-16a%28n%2B1%29+%2B2an%2B5%3D0+%2C%E8%AE%B0bn%3D1%2Fan-0.5%EF%BC%88n%E5%A4%A7%E4%BA%8E%E7%AD%89%E4%BA%8E1%EF%BC%89%E6%B1%82%E6%95%B0%E5%88%97%EF%BC%88anbn%EF%BC%89%E7%9A%84%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%E5%8F%8A%E5%89%8Dn%E9%A1%B9%E5%92%8CSn+%EF%BC%88%E6%B3%A8%EF%BC%9Aan%E4%B8%BAa%E7%9A%84%E7%AC%ACn%E9%A1%B9%2Ca%EF%BC%88n%2B1%EF%BC%89%E4%B8%BAa%E7%9A%84%E7%AC%ACn%2B1%E9%A1%B9%EF%BC%89)
数列(an)满足a1=1,且8a(n+1)an -16a(n+1) +2an+5=0 ,记bn=1/an-0.5(n大于等于1)求数列(anbn)的通项公式及前n项和Sn (注:an为a的第n项,a(n+1)为a的第n+1项)
数列(an)满足a1=1,且8a(n+1)an -16a(n+1) +2an+5=0 ,记bn=1/an-0.5(n大于等于1)求数列(anbn)的通项公式及前n项和Sn (注:an为a的第n项,a(n+1)为a的第n+1项)
数列(an)满足a1=1,且8a(n+1)an -16a(n+1) +2an+5=0 ,记bn=1/an-0.5(n大于等于1)求数列(anbn)的通项公式及前n项和Sn (注:an为a的第n项,a(n+1)为a的第n+1项)
令anbn=cn,则cn=2an/(2an-1)
下面构造2an-1,2a(n+1)-1:
c(n+1)-cn
=a(n+1)/[2a(n+1)-1]-an/(2an-1)
={(2an-1)-[2a(n+1)-1]}/{2[2a(n+1)-1](2an-1)}
=1/{2[2a(n+1)-1]}-1/[2(2an-1)]
cn=1/[2(2an-1)]
c1=0.5
8a(n+1)an -16a(n+1) +2an+5=0
2(2an-1)[2a(n+1)-1]+3(2an-1)-6[2a(n+1)-1]=0
2(2an-1)[2a(n+1)-1]=3{2[2a(n+1)-1]-(2an-1)}
1/(2an-1)-1/{2[2a(n+1)-1]}=3
2cn-c(n+1)=3
c(n+1)=2cn-3
cn=2c(n-1)-3
cn=2c(n-1)-3
=2[2c(n-2)-3]-3
=2^2c(n-2)-3-3*2
=2^2[2c(n-3)-3]-3-3*2
=2^3c(n-3)- 3(1+2+2^2)
=...
=2^(n-1)c1-3[1+2+2^2+...+2^(n-2)]
=2^(n-2)-3[2^(n-1)-1]
=3-2.5*2^(n-1)
Sn=3n-2.5*[1+2+...+2^(n-1)]
=3n-2.5*(2^n-1)
=3n-2.5*2^n+2.5