Question

So sánh

A= \(\dfrac{7^{10}}{1+7+...+7^9}\) và

B=\(\dfrac{5^{10}}{1+5+...+5^9}\)

Answer 1

Đặt :

\(C=1+7+....+7^9\) \(\Leftrightarrow A=7^{10}:C\)

\(\Leftrightarrow7C=7+7^2+.......+7^{10}\)

\(\Leftrightarrow7C-C=\left(7+7^2+......+7^{10}\right)-\left(1+7+........+7^9\right)\)

\(\Leftrightarrow6C=7^{10}-1\)

\(\Leftrightarrow C=\dfrac{7^{10}-1}{6}\)

\(\Leftrightarrow A=7^{10}:\dfrac{7^{10}-1}{6}=7^{10}.\dfrac{6}{7^{10}-1}=\dfrac{1}{6}+\dfrac{1}{\left(7^{10}-1\right)6}\)\(\left(1\right)\)

Tương tự ta có : \(B=5^{10}:\dfrac{5^{10}-1}{4}=5^{10}.\dfrac{4}{5^{10}-1}=\dfrac{1}{4}+\dfrac{1}{\left(5^{10}-1\right)4}\left(2\right)\)

Từ \(\left(1\right)+\left(2\right)\Leftrightarrow A< B\)