Tính nhanh (1+1999/1)(1+1999/2)…..(1+1999/1000)/(1+1000/1)(1+1000/2)…(1+1000/1999)

Đặt `A={(1+1999/1)(1+1999/2)…..(1+1999/1000)}/{(1+1000/1)(1+1000/2)…(1+1000/1999)}`

Tử số `A=(1+1999/1)(1+1999/2)…..(1+1999/1000)`

`={1+1999}/1.{2+1999}/2…{1000+1999}/1000`

`=2000/1 . 2001/2…2999/1000`

`={2000.2001.2999}/{1.2.3…1000}`

`={2000.2001.2999}/{1000!}`

Mẫu số `A=(1+1000/1)(1+1000/2)…..(1+1000/1000)`

`={1+1000}/1.{2+1000}/2…{1000+1000}/1000`

`=1001/1 . 1002/2…2000/1000`

`={1001.1002…2000}/{1.2.3…1000}`

`={1001.1002…2000}/{1000!}`

`⇒A={{2000.2001.2999}/{1000!}}/{{1001.1002…2000}/{1000!}}`

`A={2000.2001.2999}/{1001.1002…2000}`

`A={2001.2002.2999}/{1001.1002…1999}`