Question

1/x(x+1)(x+2)+1/(x+1)(x+2)(x+3).+1/(x+98)(x+99)(x+100)=?

Answer 1

1/x(x+1)(x+2)+1/(x+1)(x+2)(x+3).+1/(x+98)(x+99)(x+100)

=(1/2){[1/x(x+1) - 1/(x+1)(x+2)] + [1/(x+1)(x+2) - 1/(x+2)(x+3)]+.

+[1/(x+98)(x+99) - 1/(x+99)(x+100)]}

=(1/2){1/x(x+1) - 1/(x+99)(x+100)}

=[(x+99)(x+100) - x(x+1)]/2x(x+1)(x+99)(x+100)

=[198x+9900]/2x(x+1)(x+99)(x+100)

=[99x+4950]/x(x+1)(x+99)(x+100)

=99(x+50)/x(x+1)(x+99)(x+100)