数列{An}满足a1=3/2,a (n+1 )=a(n)*2-a(n)+1,则m=1/a1+1/a2+1/a3+.+1/
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数列{An}满足a1=3/2,a (n+1 )=a(n)*2-a(n)+1,则m=1/a1+1/a2+1/a3+.+1/a2011的整数部分是多少?
![数列{An}满足a1=3/2,a (n+1 )=a(n)*2-a(n)+1,则m=1/a1+1/a2+1/a3+.+1/](/uploads/image/z/20277096-24-6.jpg?t=%E6%95%B0%E5%88%97%7BAn%7D%E6%BB%A1%E8%B6%B3a1%3D3%2F2%2Ca+%28n%2B1+%29%3Da%28n%29%2A2-a%28n%29%2B1%2C%E5%88%99m%3D1%2Fa1%2B1%2Fa2%2B1%2Fa3%2B.%2B1%2F)
a (n+1 )=a(n)*2-a(n)+1
得a(n+1)-1=an(an-1)
1/[a(n+1)-1]=1/(an-1)-1/an
得1/an=1/(an-1)-1/[a(n+1)-1]
所以m=1/a1+1/a2+1/a3+.+1/a2011
=1/(a1-1)-1/(a2-1)+1/(a2-1)-1/(a3-1)+……+1/(a2011-1)-1/(a2012-1)
=1/(a1-1)-1/(a2012-1)
=2-1/(2012-1)
由于从a3开始an就大于2,所以a2012-1>1故1/(a2012-1)
得a(n+1)-1=an(an-1)
1/[a(n+1)-1]=1/(an-1)-1/an
得1/an=1/(an-1)-1/[a(n+1)-1]
所以m=1/a1+1/a2+1/a3+.+1/a2011
=1/(a1-1)-1/(a2-1)+1/(a2-1)-1/(a3-1)+……+1/(a2011-1)-1/(a2012-1)
=1/(a1-1)-1/(a2012-1)
=2-1/(2012-1)
由于从a3开始an就大于2,所以a2012-1>1故1/(a2012-1)
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