Question

六分之一+十二分之一+二十分之一+三十分之一…+2005×2006分之1

Answer 1

解：
六分之一+十二分之一+二十分之一+三十分之一…+2005×2006分之1
=1/(2×3)+1/(3×4)+1/(4×5)+1/(5×6)+……+1/(2005×2006)
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)……+(1/2005-1/2006)
=1/2-(1/3-1/3)-(1/4-1/4)-(1/5-1/5)-(1/6-1/6)……-(1/2005-1/2005)-1/2006
=1/2-1/2006
1003/2006-1/2006
=1002/2006
=501/1003

Answer 2

6=2*3 1/6=1/2-1/3
12=3*4 1/12=1/3-1/4
20=4*5 1/20=1/4-1/5……
然后列项求和
=1/2-1/2006=501/1003

Answer 3

六分之一+十二分之一+二十分之一+三十分之一…+2005×2006分之1
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+……+1/2005-1/2006
=1/2-1/2006
=1002/2006
=501/1003

Answer 4

六分之一+十二分之一+二十分之一+三十分之一…+2005×2006分之1
=(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+……+（1/2005-1/2006）
=1/2-1/3+1/3-1/4+1/4-1/5+……+1/2005-1/2006
=1/2+（-1/3+1/3）+（-1/4+1/4）+（-1/5+1/5）+……+（-1/2005+1/2005）-1/2006
=1/2-1/2006
=1002/2006
=501/1003

Answer 5

501/1003
裂项相消