求一道分数数学题.1/1*3+1/2*4+1/3*5+1/4*6+……+1/100*102=?请大家配上说明,不然写一大堆算式,我也不懂.

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求一道分数数学题.
1/1*3+1/2*4+1/3*5+1/4*6+……+1/100*102=?
请大家配上说明,不然写一大堆算式,我也不懂.

说明:1/1*3=1/2*(1/1-1/3 ) 1/2*4=1/2*(1/2-1/4)……其它各项依此类推.各项相加时提取公因数1/2
所以
1/1*3+1/2*4+1/3*5+1/4*6+……+1/100*102
=(1/1*3+1/3*5+1/5*7+1/7*9+……1/99*101)+(1/2*4+1/4*6+1/6*8+……+1/100*102)
=1/2(1-1/3+1/3-1/5+1/5-1/7+……+1/99-1/101)+1/2(1/2-1/4+1/4-1/6+1/6-1/8+……+1/100-1/102)
=1/2×(1-1/101)+1/2(1/2-1/102)
=1/2×(1-1/101+1/2-1/102)
=1/2×(100/101+50/102)
=1/2×(10200+5050)/(101×102)
=1/2×15250/10302
=7625/10302
懂了吗?