\(A=\dfrac{1+7+7^2+7^3+7^4+...+7^{2013}}{1-7^{2014}}\)

\(\Rightarrow7A=\dfrac{7+7^2+7^3+7^4+7^5+...+7^{2014}}{1-7^{2014}}\)

\(\Leftrightarrow7A-A=\dfrac{\left(7+7^2+7^3+...+7^{2014}\right)-\left(1+7+7^2+...+7^{2013}\right)}{1-7^{2014}}\) \(\Leftrightarrow6A=\dfrac{7+7^2+7^3+...+7^{2014}-1-7-7^2-....-7^{2013}}{1-7^{2014}}\) \(\Leftrightarrow6A=\dfrac{7^{2014}-1}{1-7^{2014}}\)

\(\Leftrightarrow A=\dfrac{7^{2014}-1}{1-7^{2014}}.\dfrac{1}{6}\)

\(\Rightarrow A=\dfrac{7^{2014}-1}{6-6.7^{2014}}\)

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A = (7+7^2+7^3)+(7^4+7^5+7^6)+.....(+7^2011+7^2012+7^2013)

   = (7+7^2+7^3)+7^3.(7+7^2+7^3)+....+7^2010.(7+7^2+7^3)

   = 399 + 7^3.399 + .... + 7^2010 . 399

   = 399.(1+7^3+....+7^2010) chia hết cho 399

Mà 399 chia hết cho 19 => A chia hết cho 19

a: \(=\left[-15+2013-36\right]:\left(-2\right)-1+27\)

\(=\dfrac{1962}{-2}+26\)

\(=-981+26=-955\)

b: \(=\dfrac{-35}{-7}+\left(-12\right)-\left(-1\right)=5+1-12=-6\)

c: \(=-2^9\cdot\left(-2\right)^5+6^2-1-\left(-125\right)\)

\(=2^{14}+36-1+125\)

\(=16384+36+124=16544\)

Ta có: \(\left\{{}\begin{matrix}7A=\dfrac{7^{2013}+7}{7^{2013}+1}=1+\dfrac{6}{7^{2013}+1}\\7B=\dfrac{7^{2014}+7}{7^{2014}+1}=1+\dfrac{6}{7^{2014}+1}\end{matrix}\right.\Leftrightarrow7A>7B\Leftrightarrow A>B\)