Question

So sánh
\(A=\frac{2^{2008}-3}{2^{2007}-1}\) và \(B=\frac{2^{2007}-3}{2^{2006}-1}\)

 

Answer 1

Ta có: \(\frac{2^{2008}-3}{2^{2007}-1}=\frac{\left(2^{2008}-2\right)-1}{2^{2007}-1}=\frac{2\left(2^{2007}-1\right)-1}{2^{2007}-1}=2-\frac{1}{2^{2007}-1}\)

CMTT ta có \(\frac{2^{2007}-3}{2^{2006}-1}=2-\frac{1}{2^{2006}-1}\)

MÀ 2²⁰⁰⁶-1<2²⁰⁰⁷-1 => \(\frac{1}{2^{2006}-1}>\frac{1}{2^{2007}-1}\Rightarrow2-\frac{1}{2^{2006}-1}< 2-\frac{1}{2^{2007}-1}\)

Từ đó \(\Rightarrow\frac{2^{2008}-3}{2^{2007}-1}>\frac{2^{2007}-3}{2^{2006}-1}\)