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已知2/1×3=1-1/3,2/3×5=1/3-1/5,2/5×7=1/5-1/7...利用上面的规律计算:1/1×4+1/4×7+1/7×10+...+1/301×304
题目详情
已知2/1×3=1-1/3,2/3×5=1/3-1/5,2/5×7=1/5-1/7...利用上面的规律计算:1/1×4+1/4×7+1/7×10+...+1/301×304
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答案和解析
1/(1×4)+1/(4×7)+1/(7×10)+……+1/(301×304)
=(1/3)×3/(1×4)+(1/3)×3/(4×7)+(1/3)×3/(7×10)+……+(1/3)×3/(301×304)
=(1/3)×[(1-1/4)+(1/4-1/7)+(1/7-1/10)+……+(1/301-1/304)]
=(1/3)(1-1/4+1/4-1/7+1/7-1/10+……+1/301-1/304)
=(1/3)×(1-1/304)
=(1/3)×303/304
=101/304
=(1/3)×3/(1×4)+(1/3)×3/(4×7)+(1/3)×3/(7×10)+……+(1/3)×3/(301×304)
=(1/3)×[(1-1/4)+(1/4-1/7)+(1/7-1/10)+……+(1/301-1/304)]
=(1/3)(1-1/4+1/4-1/7+1/7-1/10+……+1/301-1/304)
=(1/3)×(1-1/304)
=(1/3)×303/304
=101/304
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