Question

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,……

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)

Answer 1

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)

Answer 2

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

Answer 3

=1/(x-3)-1/x

Answer 4

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)]