Question

数学题 1×1/2+1/2×1/3+1/3×1/4+.....+1/2009×1/2010 简便方法

Answer 1

对于自然数n有：1/n * 1/(n+1)= 1/n - 1/(n+1)，所以原式等于：
1-1/2 + 1/2- 1/3 +1/3- 1/4 +........- 1/2009 + 1/2009 - 1/2010
= 1- 1/2010
=2009/2010.

Answer 2

1×1/2+1/2×1/3+1/3×1/4+.....+1/2009×1/2010
= 1/2+1/(2*3)+1/(3*4)…1/(2009*2010)
因为：
1/(2*3)=1/6=(1/2)-(1/3)
1/(3*4)=1/12=(1/3)-(1/4)
所以1/2+1/(2*3)+1/(3*4)…1/(2009*2010)
=(1/2)+(1/2)-(1/3)+(1/3)-(1/4)+……+(1/2008)-(1/2009)+(1/2009)-(1/2010)
=1-(1/2010)
=2009/2010

Answer 3

1×1/2+1/2×1/3+1/3×1/4+.....+1/2009×1/2010
=1-1/2+1/2-1/3+1/3-1/4+。。。+1/2009-1/2010
=1-1/2010
=2009/2010

Answer 4

=(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+(1/2009-1/2010|)=2009/2010