Question

Cho \(A=-\dfrac{10^{2011}+1}{10^{2012}+1}\) và \(B=-\dfrac{10^{2012}+1}{10^{2013}+1}.CMR:A< B\)

Answer 1

Trước hết ta so sánh 10.A với 10.B từ đó ta \(\Rightarrow\) A < B. Ta có:

\(10.A=\dfrac{-10\left(10^{2011}+1\right)}{10^{2012}+1}=\dfrac{-\left(10^{2011}.10+10\right)}{10^{2012}+1}\)

\(=\dfrac{-\left(10^{2012}+10\right)}{10^{2012}+1}=\dfrac{-\left(10^{2012}+1\right)}{10^{2012}+1}-\dfrac{9}{10^{2012}+1}=-1-\dfrac{9}{10^{2012}+1}\)

Tương tự: \(10.B=-1-\dfrac{9}{10^{2013}+1}\)

Do \(10^{2013}+1>10^{2012}+1,\) nên \(\dfrac{-9}{10^{2013}+1}>\dfrac{-9}{10^{2012}+1}\)

Do đó \(10.A< 10.B,\) tức là \(A< B\)