1⼀1*2*3+1⼀2*3*4+.....+1⼀98*99*100=?

2 3 99 1 2 98
----- + ------+...+ --------- -( -------- + ---------- + ----------)
1*2*3 2*3*4 98*99*100 1*2*3 2*3*4 98*99*100

=1/2(1/1-1/3+1/2-1/4+....+1/98-1/100)- (1/2-1/3+1/3-1/4+...+1/99-1/100)

=1/2(1+1/2-1/99-1/100)-(1/2-1/100)

=1/4-1/198+1/200

=4949/19800

2/1*2*3 = 1/1*2 - 1/2*3
2/2*3*4 = 1/2*3 - 1/3*4

原式 = 1/2 * [1/1*2 - 1/2*3 + 1/2*3 - 1/3*4 + ..... - 1/99*100] =
1/2 * [ 1/1*2 - 1/99*100 ] =
1/2 * 4949/9900 =
4949/19800