有理数的巧算,1.计算:1/2+1/4+1/8+...+1/1024提示:1/4=1/2*1/2,1/8=1/2*1/4...因此,本题运用错位相减法教简便2.计算:1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)
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![有理数的巧算,1.计算:1/2+1/4+1/8+...+1/1024提示:1/4=1/2*1/2,1/8=1/2*1/4...因此,本题运用错位相减法教简便2.计算:1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)](/uploads/image/z/6964647-15-7.jpg?t=%E6%9C%89%E7%90%86%E6%95%B0%E7%9A%84%E5%B7%A7%E7%AE%97%2C1.%E8%AE%A1%E7%AE%97%EF%BC%9A1%2F2%2B1%2F4%2B1%2F8%2B...%2B1%2F1024%E6%8F%90%E7%A4%BA%EF%BC%9A1%2F4%3D1%2F2%2A1%2F2%2C1%2F8%3D1%2F2%2A1%2F4...%E5%9B%A0%E6%AD%A4%2C%E6%9C%AC%E9%A2%98%E8%BF%90%E7%94%A8%E9%94%99%E4%BD%8D%E7%9B%B8%E5%87%8F%E6%B3%95%E6%95%99%E7%AE%80%E4%BE%BF2.%E8%AE%A1%E7%AE%97%EF%BC%9A1%2F2%2B%281%2F3%2B2%2F3%29%2B%281%2F4%2B2%2F4%2B3%2F4%29%2B%281%2F5%2B2%2F5%2B3%2F5%2B4%2F5%29%2B...%2B%281%2F60%2B2%2F60%2B3%2F60%2B...%2B59%2F60%29)
有理数的巧算,1.计算:1/2+1/4+1/8+...+1/1024提示:1/4=1/2*1/2,1/8=1/2*1/4...因此,本题运用错位相减法教简便2.计算:1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)
有理数的巧算,
1.计算:1/2+1/4+1/8+...+1/1024
提示:1/4=1/2*1/2,1/8=1/2*1/4...因此,本题运用错位相减法教简便
2.计算:1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)
有理数的巧算,1.计算:1/2+1/4+1/8+...+1/1024提示:1/4=1/2*1/2,1/8=1/2*1/4...因此,本题运用错位相减法教简便2.计算:1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)
1/2+1/4+1/8+...+1/1024
再加上一个1/1024,然后减去一个1024,得:
原式=(1/2+1/4+1/8+...+1/1024 +1/1024)-1/1024
因为:1/2^n+1/2^n=2*1/2^n=1/2^(n-1)
【注:2^n意为2的n次方】
所以原式=1/2+1/4+1/8+...+1/512+1/512-1/1024
=1/2+1/4+1/8+...+1/256+1/256-1/1024
=...
=1-1/1024
=1023/1024
观察括号内的东西:可以得知:
括号内的通项为:
1/(n+1)+2/(n+1)+...+n/(n+1)
=(1+2+3+...+n)/(n+1)
由等差数列求和公式:1+2+..+n=n(n+1)/2,代回得:
=[n(n+1)/2]/(n+1)
=n/2
所以:
1/2+(1/3+2/3)+(1/4+2/4+3/4)+(1/5+2/5+3/5+4/5)+...+(1/60+2/60+3/60+...+59/60)
=1/2+2/2+3/2+...+59/2 【一共59项】
=(1+2+...+59)/2
=885
希望我的回答让你满意