计算:1/1*3+1/2*4+1/3*5+…+1/2011*2013+1/2012*2014
问题描述：
计算:1/1*3+1/2*4+1/3*5+…+1/2011*2013+1/2012*2014
答
原式=1/2(1-1/3+1/3-1/5+...+/2011-1/2013)+1/2(1/2-1/4+1/4-1/6+...+1/2012-1/2014)
=1/2(1-1/2013)+1/2(1/2-1/2014)
=1/2×2012/2013+1/2×1006/2014
=1006/2013+503/2014
=3038623/4054182
答
裂项
1/1*3+1/2*4+1/3*5+…+1/2011*2013+1/2012*2014
=(1/2)(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+.......+1/2011-1/2013+1/2012-1/2014)
=(1/2)(1+1/2-1/2013-1/2014)
=(1/2)(3/2-1/2013-1/2014)
答
原式=1/2(1-1/3+1/3-1/5+...+/2011-1/2013)+1/2(1/2-1/4+1/4-1/6+...+1/2012-1/2014)=1/2(1-1/2013)+1/2(1/2-1/2014)=1/2×2012/2013+1/2×1006/2014=1006/2013+503/2014=3038623/4054182