一道数学培优题(日期09.11-1) 类似题的规律请说明第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999第三题:1/18+1/54+1/108+1/180+1/270
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![一道数学培优题(日期09.11-1) 类似题的规律请说明第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999第三题:1/18+1/54+1/108+1/180+1/270](/uploads/image/z/11522499-51-9.jpg?t=%E4%B8%80%E9%81%93%E6%95%B0%E5%AD%A6%E5%9F%B9%E4%BC%98%E9%A2%98%EF%BC%88%E6%97%A5%E6%9C%9F09.11-1%EF%BC%89+%E7%B1%BB%E4%BC%BC%E9%A2%98%E7%9A%84%E8%A7%84%E5%BE%8B%E8%AF%B7%E8%AF%B4%E6%98%8E%E7%AC%AC%E4%B8%80%E9%A2%98%EF%BC%9A1%2F2%2B1%2F4%2B1%2F8%2B1%2F16%2B1%2F32%2B1%2F64%2B1%2F128%2B1%2F256%2B1%2F512%2B1%2F1024%E7%AC%AC%E4%BA%8C%E9%A2%98%EF%BC%9A1%2F1%C3%973%2B1%2F3%C3%975%2B1%2F5%C3%977%2B%E2%80%A6%E2%80%A6%2B1%2F1997%C3%971999%E7%AC%AC%E4%B8%89%E9%A2%98%EF%BC%9A1%2F18%2B1%2F54%2B1%2F108%2B1%2F180%2B1%2F270)
一道数学培优题(日期09.11-1) 类似题的规律请说明第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999第三题:1/18+1/54+1/108+1/180+1/270
一道数学培优题(日期09.11-1) 类似题的规律请说明
第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999
第三题:1/18+1/54+1/108+1/180+1/270
一道数学培优题(日期09.11-1) 类似题的规律请说明第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999第三题:1/18+1/54+1/108+1/180+1/270
1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
=(1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024+1/1024)-1/1024
=1-1/1024
=1023/1024
方法总结:添项,倒推法,1/1024+1/1024=1/512,1/512+1/512=1/256……
1/1×3+1/3×5+1/5×7+……+1/1997×1999
=(1-1/3+1/3-1/5+1/5-1/7+……+1/1997-1/1999)/2
=(1-1/1999)/2
=999/1999
1/18+1/54+1/108+1/180+1/270
=1/3*6+1/6*9+1/9*12+1/12*15+1/15*18
=(1/3-1/6+1/6-1/9+……+1/15-1/18)/3
=(1/3-1/18)/3
=5/54
2.3两题方法一样,就是裂项法,一个分数拆成2个分数差的形式,从而得到前后相邻项的抵消,达到化简的目的
第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
先加上一个1/1024,然后再减去一个1/1024
1/1024+1/1024=1/512
1/512+1/512=1/256
1/26+1/256=1/128
1/128+1/128=1/64
.....
1/4+1/4=1/2<...
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第一题:1/2+1/4+1/8+1/16+1/32+1/64+1/128+1/256+1/512+1/1024
先加上一个1/1024,然后再减去一个1/1024
1/1024+1/1024=1/512
1/512+1/512=1/256
1/26+1/256=1/128
1/128+1/128=1/64
.....
1/4+1/4=1/2
1/2+1/2=1
原式=1-1/1024=1023/1024
第二题:1/1×3+1/3×5+1/5×7+……+1/1997×1999
1/(1×3)=1/2x(1-1/3)
1/(3×5)=1/2x(1/3-1/5)
......
1/(1997×1999)=1/2x(1/1997-1/1999)
原式=1/2x(1-1/3+1/3-1/5+1/5-1/7+....+1/1997-1/1999)
=1/2x(1-1/1999)
=1/2x(1998/1999)
=999/1999
第三题:1/18+1/54+1/108+1/180+1/270
=1/18+1/18x1/3+1/18x1/6+1/18x1/10+1/18x1/15
=1/18x(1+1/3+1/6+1/10+1/15)
=1/18X5/3
=5/54
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