Question

1×3分之1+3×5分之1+5×7分之1+……+17×19分之1

1×11分之1+11×21分之1+21×31+……+2001×2011分之1

Answer 1

1×11分之1+11×21分之1+21×31+……+2001×2011分之1

=1/(1×11)+1/(11×21)+.....+1/(2001×2011)
=(1/10）×(1-1/11)+(1/10）×(1/11-1/21)+....+(1/10）×(1/2001-1/2011)
=(1/10)×(1-1/11+1/11-1/21+1/21-1/31+....+1/2001-1/2011)
=(1/10)×(1-1/2011)
=(1/10)×2000/2011
=200/2011
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Answer 2

1/1×3+1/3×5+1/5×7+……+1/17×19
=1/2(1/1-1/3+1/3-1/5+1/5-1/7+……+1/17-1/19)
=1/2(1-1/19)
=1/2×18/19
=9/19

1/1×11+1/11×21+1/21×31+……+1/2001×2011

=1/10(1/1-1/11+1/11-1/21+1/21-1/31+……+1/2001-1/2011)
=1/10(1-1/2011)
=1/10×2010/2011
=201/2011