Question

x(x+1)分之一+(x+1)(x+2)分之一+(x+2)(x+3)分之一+(x+3)(x+4)分之一+(x+4)(x+5)分之一

用简便方法

Answer 1

1/[x(x+1)]+1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]+1/[(x+4)(x+5)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+1/(x+4)-1/(x+5)
=1/x-1/(x+5)
=(x+5-x)/[x(x+5)]
=5/(x^2+5x)

Answer 2

1/[x(x+1)]+1/[(x+1)(x+2)]+1/[(x+2)(x+3)]+1/[(x+3)(x+4)]+1/[(x+4)(x+5)]
=1/x-1/(x+1)+1/(x+1)-1/(x+2)+1/(x+2)-1/(x+3)+1/(x+3)-1/(x+4)+1/(x+4)-1/(x+5)
=1/x-1/(x+5)
=(x+5-x)/[x(x+5)]
=5/(x^2+5x)