奥数 1⼀2*3*4+1⼀3*4*5+..............1⼀2001*2002*2003 的答案,题解,公式中文和英文)

1/(2*3*4)+1/(3*4*5)+..............1/(2001*2002*2003)
=1/2*[1/(2*3)-1/(3*4)]+1/2*[1/(3*4)-1/(4*5)]+……+1/2*[1/(2001*2002)-1/(2002*2003)]
=1/2*[1/(2*3)-1/(3*4)+1/(3*4)-1/(4*5)+1/(4*5)+……+1/(2001*2002)-1/(2002*2003)]
=1/2*[1/6-1/(2002*2003)]
=1/2*[1/6-1/4010006]
=1/2*(4010006-6)/24060036
=4010000/48120072
=501250/6015009 .

1/(2*3*4)=3/24-(1/3-1/4)=1/(2*4)-(1/3-1/4)=(1/2-1/4)/2-(1/3-1/4)
1/(3*4*5)=4/60-(1/4-1/5)=1/(3*5)-(1/4-1/5)=(1/3-1/5)/2-(1/4-1/5)
:
:
1/(2001*2002*2003)=(1/2001-1/2003)/2-(1/2002-1/2003)
所以题中式子
=(1/2+1/3-1/2002-1/2003)/2-(1/3-1/2003)
=1/12-1/4004+1/4006
=1/12-1/(2002*4006)

1/2*3*4+1/3*4*5+...+1/2001*2002*2003
=(1/2)[(1/2*3-1/3*4)+(1/3*4-1/4*5)+(1/4*5-1/5*6)+...+(1/2001*2002-1/2002*2003)]
=(1/2)[1/2*3-1/2002*2003]
=1002500/12030018

1/2*3*4=3/2*3*3*4=3（1/8*9）
1/（n-1）*n*(n+1)=n(1/n*n*(n*n-1))=n(1/(n*n-1)-1/n*n)=n/(n-1)(n+1)-1/n