So sánh A và B biết: A= $\frac{10^{15}+1 }{10^{16}+1}$ và B=$\frac{10^{16}+1 }{10^{17}+1}$ Nhanh nha :) 18/08/2021 Bởi Eva So sánh A và B biết: A= $\frac{10^{15}+1 }{10^{16}+1}$ và B=$\frac{10^{16}+1 }{10^{17}+1}$ Nhanh nha 🙂
$ 10A = \dfrac{10^{16} +10}{10^{16}+1} = \dfrac{(10^{16} +1)+9}{10^{16}+1}$ $ = \dfrac{10^{16} +1}{10^{16}+1} + \dfrac{9}{10^{16}+1} = 1 + \dfrac{9}{10^{16}+1}$ $ 10B = \dfrac{10^{17} +10}{10^{17}+1} = \dfrac{(10^{17} +1)+9}{10^{17}+1}$ $ = \dfrac{10^{17} +1}{10^{17}+1} + \dfrac{9}{10^{17}+1} = 1 + \dfrac{9}{10^{17}+1}$ Ta có $ 10^{16}+1 < 10^{17}+1 \to \dfrac{9}{10^{16}+1} > \dfrac{9}{10^{17}+1}$ $\to 1 + \dfrac{9}{10^{16}+1} > \dfrac{9}{10^{17}+1}$ $\to 10A > 10B$ $\to A > B$ Bình luận
`B= (10^16+1) / (10^17 + 1)` `⇒ B < (10^16+1+9) / (10^17 + 1 +9)` `= (10^16+10) / (10^17 + 10)` `=(10.(10^15+1))/(10.(10^16+1))` `=(10^15+1) / (10^16 + 1)` `=A` `⇒B=A` Bình luận
$ 10A = \dfrac{10^{16} +10}{10^{16}+1} = \dfrac{(10^{16} +1)+9}{10^{16}+1}$
$ = \dfrac{10^{16} +1}{10^{16}+1} + \dfrac{9}{10^{16}+1} = 1 + \dfrac{9}{10^{16}+1}$
$ 10B = \dfrac{10^{17} +10}{10^{17}+1} = \dfrac{(10^{17} +1)+9}{10^{17}+1}$
$ = \dfrac{10^{17} +1}{10^{17}+1} + \dfrac{9}{10^{17}+1} = 1 + \dfrac{9}{10^{17}+1}$
Ta có $ 10^{16}+1 < 10^{17}+1 \to \dfrac{9}{10^{16}+1} > \dfrac{9}{10^{17}+1}$
$\to 1 + \dfrac{9}{10^{16}+1} > \dfrac{9}{10^{17}+1}$
$\to 10A > 10B$
$\to A > B$
`B= (10^16+1) / (10^17 + 1)`
`⇒ B < (10^16+1+9) / (10^17 + 1 +9)`
`= (10^16+10) / (10^17 + 10)`
`=(10.(10^15+1))/(10.(10^16+1))`
`=(10^15+1) / (10^16 + 1)`
`=A`
`⇒B=A`