Toán So sánh: $A=\frac{2^{2010}+1}{2^{2007}+1}, B=\frac{2^{2012}+1}{2^{2009}+1}$ 13/07/2021 By aikhanh So sánh: $A=\frac{2^{2010}+1}{2^{2007}+1}, B=\frac{2^{2012}+1}{2^{2009}+1}$
Đáp án: $A < B$ Giải thích các bước giải: Ta có: $A = \dfrac{2^{2010} + 1}{2^{2007} + 1}$ $= \dfrac{2^{2007}.8 + 1}{2^{2007} + 1}$ $= \dfrac{8(2^{2007} + 1) – 7}{2^{2007} + 1}$ $= 8 – \dfrac{7}{2^{2007} +1}$ Tương tự: $B = 8 – \dfrac{7}{2^{2009} + 1}$ Mặt khác: $2^{2007} + 1 < 2^{2009} + 1$ $\Leftrightarrow \dfrac{7}{2^{2007} + 1} > \dfrac{7}{2^{2009} + 1}$ $\Leftrightarrow – \dfrac{7}{2^{2007} + 1} < -\dfrac{7}{2^{2009} + 1}$ $\Leftrightarrow 8 – \dfrac{7}{2^{2007} + 1} < 8 -\dfrac{7}{2^{2009} + 1}$ Hay $A < B$ Trả lời
`A=(2^{2010}+1)/(2^{2007}+1)` `⇒A/8=(2^{2010}+1)/(2^{2010}+8)` `⇒A/8=((2^{2010}+8)-7)/(2^{2010}+8)` `⇒A/8=1-7/(2^{2010}+8)` `B=(2^{2012}+1)/(2^{2009}+1)` `⇒B/8=(2^{2012}+1)/(2^{2012}+8)` `⇒B/8=((2^{2012}+8)-7)/(2^{2012}+8)` `⇒B/8=1-7/(2^{2012}+8)` Vì `2^{2010}+8<2^{2012}+8` `⇒7/(2^{2010}+8)>7/(2^{2012}+8)` `⇒1-7/(2^{2010}+8)<1-7/(2^{2012}+8)` `⇒A/8<B/8⇒A<B` Vậy $A<B$. Trả lời
Đáp án:
$A < B$
Giải thích các bước giải:
Ta có:
$A = \dfrac{2^{2010} + 1}{2^{2007} + 1}$
$= \dfrac{2^{2007}.8 + 1}{2^{2007} + 1}$
$= \dfrac{8(2^{2007} + 1) – 7}{2^{2007} + 1}$
$= 8 – \dfrac{7}{2^{2007} +1}$
Tương tự:
$B = 8 – \dfrac{7}{2^{2009} + 1}$
Mặt khác:
$2^{2007} + 1 < 2^{2009} + 1$
$\Leftrightarrow \dfrac{7}{2^{2007} + 1} > \dfrac{7}{2^{2009} + 1}$
$\Leftrightarrow – \dfrac{7}{2^{2007} + 1} < -\dfrac{7}{2^{2009} + 1}$
$\Leftrightarrow 8 – \dfrac{7}{2^{2007} + 1} < 8 -\dfrac{7}{2^{2009} + 1}$
Hay $A < B$
`A=(2^{2010}+1)/(2^{2007}+1)`
`⇒A/8=(2^{2010}+1)/(2^{2010}+8)`
`⇒A/8=((2^{2010}+8)-7)/(2^{2010}+8)`
`⇒A/8=1-7/(2^{2010}+8)`
`B=(2^{2012}+1)/(2^{2009}+1)`
`⇒B/8=(2^{2012}+1)/(2^{2012}+8)`
`⇒B/8=((2^{2012}+8)-7)/(2^{2012}+8)`
`⇒B/8=1-7/(2^{2012}+8)`
Vì `2^{2010}+8<2^{2012}+8`
`⇒7/(2^{2010}+8)>7/(2^{2012}+8)`
`⇒1-7/(2^{2010}+8)<1-7/(2^{2012}+8)`
`⇒A/8<B/8⇒A<B`
Vậy $A<B$.