A=\(\dfrac{7^{10}}{1+7+7^2+...+7^9}\)nên \(\dfrac{1}{A}=\dfrac{1+7+7^2+...+7^9}{7^{10}}\)
\(=\dfrac{1}{7^{10}}+\dfrac{7}{7^{10}}+\dfrac{7^2}{7^{10}}+...+\dfrac{7^9}{7^{10}}\)\(=\dfrac{1}{7^{10}}+\dfrac{1}{7^9}+\dfrac{1}{7^8}+...+\dfrac{1}{7}\)
B=\(\dfrac{5^{10}}{1+5+5^2+...+5^9}\)nên \(\dfrac{1}{B}=\dfrac{1+5+5^2+...+5^9}{5^{10}}\)
\(=\dfrac{1}{5^{10}}+\dfrac{5}{5^{10}}+\dfrac{5^2}{5^{10}}+...+\dfrac{5^9}{5^{10}}\)\(=\dfrac{1}{5^{10}}+\dfrac{1}{5^9}+\dfrac{1}{5^8}+...+\dfrac{1}{5^7}\)
Ta thấy:\(\dfrac{1}{7^{10}}< \dfrac{1}{5^{10}};\dfrac{1}{7^9}< \dfrac{1}{5^9};...;\dfrac{1}{7}< \dfrac{1}{5}\) nên \(\dfrac{1}{A}< \dfrac{1}{B}\)
Vậy A<B
