Question

So sánh: P=\(\frac{10^{2010}-1}{10^{2011}-1}\) và Q=\(\frac{10^{2009}-1}{10^{2010}-1}\)

Answer 1

\(P=\dfrac{10^{2010}-1}{10^{2011}-1}\)

\(\Rightarrow10P=\dfrac{10\left(10^{2010}-1\right)}{10^{2011}-1}=\dfrac{10^{2011}-10}{10^{2011}-1}=\dfrac{10^{2011}-1-9}{10^{2011}-1}=1-\dfrac{9}{10^{2011}-1}\)

\(Q=\dfrac{10^{2009}-1}{10^{2010}-1}\)

\(\Rightarrow10Q=\dfrac{10\left(10^{2009}-1\right)}{10^{2010}-1}=\dfrac{10^{2010}-10}{10^{2010}-1}=\dfrac{10^{2010}-1-9}{10^{2010}-1}=1-\dfrac{9}{10^{2010}-1}\)Mà\(\dfrac{9}{10^{2011}-1}< \dfrac{9}{10^{2010}-1}\)

\(\Rightarrow1-\dfrac{9}{10^{2011}-1}>1-\dfrac{9}{10^{2010}-1}\)

Hay \(10P>10Q\Rightarrow P>Q\)