一道数学题的简便运算1/1×3+1/2×4+1/3×5+…+1/10×12

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一道数学题的简便运算1/1×3+1/2×4+1/3×5+…+1/10×12

一道数学题的简便运算1/1×3+1/2×4+1/3×5+…+1/10×12
一道数学题的简便运算
1/1×3+1/2×4+1/3×5+…+1/10×12

一道数学题的简便运算1/1×3+1/2×4+1/3×5+…+1/10×12
原式=1/2(1/1*2-1/2*3)+1/2(1/2*3-1/3*4)+……+1/2(1/10*11-1/11*12)
=1/2(1/1*2-1/2*3+……+1/10*11-1/11*12)
=1/2(1/1*2-1/11*12)
=65/264

1/(1*3)=(1-1/3)*1/2
1/(2*4)=(1/2-1/4)*1/2
1/(3*5)=(1/3-1/5)*1/2
......
1/(10*12)=(1/10-1/12)*1/2
原式
=1/2*[(1-1/3)+(1/2-1/4)+(1/3-1/5)+...+(1/10-1/12)]
=1/2*(1+1/2-1/3+1/3-1/4+1/4-....-1/9+1/9+1/10-1/11+1/11-1/12)
=1/2*(1+1/2+1/10-1/12)
=1/2*91/60
=91/120

=1/2*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+……+1/8-1/10+1/9-1/11+1/10-1/12),
化减得1/2*(1+1/2-1/11-1/12)=1/2*(175/132)=175/264