Tính nhanh 1/2+5/6+11/12+19/20+29/30+41/42+55/56+71/72+89/90

Lời giải:

`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`

$=(1+1+1+1+1+1+1+1+1)-(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}-\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{9}+\dfrac{1}{9}-\dfrac{1}{10})$

`=9-(1/1 - 1/10)`

`=9-9/10`

`=81/10`

Đáp án:

` (81)/(10)`

Giải thích các bước giải:

`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/(1.2)+1/(2.3)+1/(3.4)+1/(4.5)+1/(5.6)+1/(6.7)+1/(7.8)+1/(8.9)+1/(9.10)] `

`=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)`.