分数裂项6题

第一题：3/1x2x3x4+3/2x3x4x5+……+3/17x18x19x20

=1/1x2x3-1/2X3x4+1/2x3x4-1/3x4x5+……+1/17x18x19-1/18x19x20

=1/1x2x3-1/18x19x20

=1/6-6840

=1139/6840

第二题：2/1x(1+2)+3/(1+2)x（1+2+3）+4/(1+2+3)x(1+2+3+4)+……+100/(1+2+……+99)x(1+2+……+100)

=1-1/(1+2)+1/(1+2)-1/（1+2+3）+1/(1+2+3)-1/(1+2+3+4)+……+1/(1+2+……+99)-1/(1+2+……+100)

=1-1/(1+2+……+100)

=1-1/5050

=5049/5050

第三题：3/1^2x2^2+5/2^2x3^2+7/3^2x4^2+……+15/7^2x8^2

=1/1^2-1/2^2+1/2^2-1/3^2+1/3^2-1/4^2+……+1/7^2-1/8^2

=1/1^2-1/8^2

=1-1/64

=63/64

第四题：3^2+1/3^2-1+5^2+1/5^2-1+7^2+1/7^2-1+……+1993^2+1/1993^2-1+1995^2+1/1995^2-1=1+2/3^2-1+1+2/5^2-1+1+2/7^2-1+……+1+2/1993^2-1+1+2/1995^2-1

=997+2/3^2-1+2/5^2-1+2/7^2-1+……+2/1993^2-1+2/1995^2-1

=997+2/2x4+2/4x6+2/6x8+……+2/1992x1994+2/1994x1996

=997+1/2-1/4+1/4-1/6+1/6-1/8+……+1/1992-1/1994+1/1994-1/1996

=997+1/2-1/1996

=997又997/1996

第五题：2x2/1x3+4x4/3x5+6x6/5x7+8x8/7x9+10x10/9x11

=1+1/1x3+1+1/3x5+1+1/5x7+1+1/7x9+1+1/9x11

=5+1/1x3+1/3x5+1/5x7+1/7x9+1/9x11

=5+1/2x(1-1/3+1/3-1/5+1/5-1/7+1/7-1/9+1/9-1/11)

=5+1/2x(1-1/11)

=5+5/11

=5又5/11

第六题:1+1/1+2+1/1+2+3+……+1/1+2+……2007

=2/2+2/6+2/12+……+2/2007x2008

=2x(1-1/2+1/2-1/3+1/3-1/4+……+1/2007-1/2008）

=2x(1-1/2008)

=2-1/1004

=1又1003/1004