a与c计算公式 计算1/{a（a+1）}+1/{（a+1)(a+1)}+{(a+2)(a+3)}+.+1/{(a+2005)(a+2006))}

计算1/{a（a+1）}+1/{（a+1)(a+1)}+{(a+2)(a+3)}+.+1/{(a+2005)(a+2006))}  

计算1/{a（a+1）}+1/{（a+1)(a+1)}+{(a+2)(a+3)}+.....+1/{(a+2005)(a+2006))}

1/{a（a+1）}+1/{（a+1)(a+1)}+{(a+2)(a+3)}+.....+1/{(a+2005)(a+2006))}
=1/a-1/(a+1）+1/（a+1)-1/(a+1)}++1/(a+2)-1/(a+3)+.....+1/(a+2005)-1/(a+2006))
=1/a-1/(a+2006))
=2006/[a(a+2006)]

计算1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+···+1/(a+2004)(a+2005)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+---+1/(a+2004)-1/(a+2005)
=1/a-1/(a+2005)
=2005/(a²+2005a)

计算:1/[a(a+1)]+1/[(a+1)(a+2)]+1/[(a+2)(a+3)+...+1/[(a+2006)(a+2007)]

1/[a(a+1)]=1/a-1/(a+1)
1/[(a+1)(a+2)]=1/(a+1)-1/(a+2)
1/[(a+2)(a+3)]=1/(a+2)-1/(a+3)
……
1/[(a+2005)(a+2006)]=1/(a+2005)-1/(a+2006)
1/[(a+2006)(a+2007)]=1/(a+2006)-1/(a+2007)
原式
1/[a(a+1)]+1/[(a+1)(a+2)]+1/[(a+2)(a+3)+...+1/[(a+2006)(a+2007)]
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+……+1/(a+2005)-1/(a+2006)+1/(a+2006)-1/(a+2007)
=1/a-1/(a+2007)
=2007/[a(a+2007)]

计算：{1/a(a+1)}+{1/（a+1)(a+2)}+{1/(a+2)(a+3)}+…+{1/(a+2006)(a+2007)}+{1/(a+2007(a+2008)}

这个，我们来分析一下，
首先，这是一个有规律可以找的，
一，分母是两个数的积，并且，相差1
二，分子，都是 1
三，这样，想到，如果，分子，分母相差1，且是积，
如1/ab，a-b=1，这时，1/ab=1/b-1/a
四，做，：
{1/a(a+1)}+{1/（a+1)(a+2)}+{1/(a+2)(a+3)}+…+{1/(a+2006)(a+2007)}+{1/(a+2007(a+2008)}
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+…+1/(a+2006)-1/(a+2007)+1/(a+2007)-1/(a+2008)
=1/a-1/(a+2008)
=2008/a(a+2008)

计算：1/a(a+1)+1/(a+1)(a+2)+...+1/(a+2006)(a+2007)

裂项：1/a(a+1)=1/a-1/(a+1)
1/a(a+1)+1/(a+1)(a+2)+...+1/(a+2006)(a+2007)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+...+1/(a+2006)-1/(a+2007)
=1/a-1/(a+2007)

1/a(a+1)+1/(a+1)(a+2)+......1/(a+2006)(a+2007)=?

用裂项公式
1/a-1/(a+1)+1/(a+1)-1/(a+2)……+1/(a+2006)-1/（a+2007)
=1/a-1/(a+2007)
=(a+2007)-a/a(a+2007)
=2007/a^2+1/a

计算1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+......+1/(a+2012)(a+2013)

原式=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+......+1/(a+2012)-1/(a+2013)
=1/a-1/(a+2013)
=2013/(a²+2013a)

计算:1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+......+1/(a+2003)(a+2004)

1/a(a+1) + 1/(a+1)(a+2) + 1/(a+2)(a+3)....1/(a+2003)(a+2004)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3).....+1/(a+2003)-1/(a+2004)
=1/a-1/(a+2004)
=2004/a(a+2004)
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计算：1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+…1/(a+2009)(a+2010)

原式 =1/a-1/(a+1)+1/(a+1)-1/(a+2)+...+1/(a+2009)-1/(a+2010) =1/a-1/(a+2010)

计算:1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+9)(a+10)

1/a(a+1)+1/(a+1)(a+2)+1/(a+2)(a+3)+...+1/(a+9)(a+10)
=1/a-1/(a+1)+1/(a+1)-1/(a+2)+1/(a+2)-1/(a+3)+...+1/(a+5)-1/(a+10)
=1/a-1/(a+10)
=10/a(a+10)