Question

Tính nhanh 5/6+11/12+19/20+29/30+41/42+55/56+71/71+89/90

Answer 1

`1/2+5/6+11/12+19/20+29/30+41/42+55/56+71/72+89/90`
`= 1-1/2+1-1/6+1-1/12+1-1/20+1-1/30+1-1/42+1-1/56+1-1/72+1-1/90`
`= 9 – (1/2+1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90)`
`= 9 – [1/(1x2)+1/(2x3)+1/(3x4)+1/(4x5)+1/(5x6)+1/(6x7)+1/(7x8)+1/(8x9)+1/(9x10)] `
`= 9 – ( 1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10) `
`= 9 – (1 – 1/10) = 9 – 9/10 = 81/10`

Answer 2

`5/6+11/12+19/20+29/30+41/42+55/56+71/72+89/90`

`=(1-1/6)+(1-1/12)+(1-1/20)+(1-1/30)+(1-1/42)+(1-1/56)+(1-1/72)+(1-1/90)`

`= 1-1/6+1-1/12+1-1/20+1-1/30+1-1/42+1-1/56+1-1/72+1-1/90`

`= (1+1+1+1+1+1+1+1)(` có `8` chữ số `1)-(1/6+1/12+1/20+1/30+1/42+1/56+1/72+1/90)`

`= 8-(1/(2xx3)+1/(3xx4)+1/(4xx5)+1/(5xx6)+1/(6xx7)+1/(7xx8)+1/(8xx9)+1/(9xx10))`

`= 8 - (1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+1/8-1/9+1/9-1/10)`

`= 8 - (1/2-1/10)`

`= 8 - 2/5`

`= 38/5`