(-1*1/2)+(-1/2*1/3)+(-1/3*1/4)+(-1/4*1/5)+""+(-1/2005*1/2006)+(-1/2006*1/2007)等于多少?

题目

(-1*1/2)+(-1/2*1/3)+(-1/3*1/4)+(-1/4*1/5)+""+(-1/2005*1/2006)+(-1/2006*1/2007)等于多少?

答案

原式=(-1)×[1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+……+1/(2005×2006)+1/(2006×2007)]
=(-1)×[(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+……+(1/2005-1/2006)+(1/2006-1/2007)]
=(-1)×(1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+……+1/2005-1/2006+1/2006-1/2007)
=(-1)×(1-1/2007)
=(-1)×2006/2007
= -2006/2007