观察下列各等式1/(1×2)=1-1/2,1/(2×3)=1/2-1/3,1/(3×4)=1/3+1/4,…… 利用规律计算1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)
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![观察下列各等式1/(1×2)=1-1/2,1/(2×3)=1/2-1/3,1/(3×4)=1/3+1/4,…… 利用规律计算1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)](/uploads/image/z/5433176-56-6.jpg?t=%E8%A7%82%E5%AF%9F%E4%B8%8B%E5%88%97%E5%90%84%E7%AD%89%E5%BC%8F1%2F%281%C3%972%29%3D1-1%2F2%2C1%2F%282%C3%973%29%3D1%2F2-1%2F3%2C1%2F%283%C3%974%29%3D1%2F3%2B1%2F4%2C%E2%80%A6%E2%80%A6+%E5%88%A9%E7%94%A8%E8%A7%84%E5%BE%8B%E8%AE%A1%E7%AE%971%2Fx%28x-1%29%2B1%2F%28x-1%29%28x-2%29%2B1%2F%28x-2%29%28x-3%29)
观察下列各等式1/(1×2)=1-1/2,1/(2×3)=1/2-1/3,1/(3×4)=1/3+1/4,…… 利用规律计算1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)
观察下列各等式1/(1×2)=1-1/2,1/(2×3)=1/2-1/3,1/(3×4)=1/3+1/4,……
利用规律计算1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)
观察下列各等式1/(1×2)=1-1/2,1/(2×3)=1/2-1/3,1/(3×4)=1/3+1/4,…… 利用规律计算1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)
裂项1/(x-1)-1/x+1/x-2-1/(x-1)...+1/(x-3)=1/(x-1)+1/(x-3)=(2x-4)/(x-1)(x-3)
=1/(x-3)-1/x
最大的公因数是1的两个自然数,叫做互质数。又是两个数是最大公因数只有1的两个数是互质数。
相邻的两个奇数是互质数。例如 49与 51。
所给示例中,分数的分母都是互质的。
原式=[1/x-1/(x-1)]+[1/(x-1)-1/(x-2)]+[1/(x-2)-1/(x-3)]
=1/x-1/(x-1)+1/(x-1)-1/(x-2)+1/(x-2)-1...
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最大的公因数是1的两个自然数,叫做互质数。又是两个数是最大公因数只有1的两个数是互质数。
相邻的两个奇数是互质数。例如 49与 51。
所给示例中,分数的分母都是互质的。
原式=[1/x-1/(x-1)]+[1/(x-1)-1/(x-2)]+[1/(x-2)-1/(x-3)]
=1/x-1/(x-1)+1/(x-1)-1/(x-2)+1/(x-2)-1/(x-3)
=1/x-1/(x-3)
=1/[x(x-3)]
收起
1/x(x-1)+1/(x-1)(x-2)+1/(x-2)(x-3)
=1/(x-1)-1/x+1/(x-2)-1/(x-1)+1/(x-3)-1/(x-2)
=1/(x-1)-1/(x-1)+1/(x-2)-1/(x-2)-1/x)+1/(x-3)
=0+0-1/x+1/(x-3)
=1/(x-3)-1/x