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\(\dfrac{x-1}{2016}+\dfrac{x-2}{2015}+\dfrac{x-3}{2014}+\dfrac{x-4}{2013}+\dfrac{x-5}{2012}-5=0\)
\(\Rightarrow\dfrac{x-1}{2016}+\dfrac{x-2}{2015}+\dfrac{x-3}{2014}+\dfrac{x-4}{2013}+\dfrac{x-5}{2012}=5\)
\(\Rightarrow\left(\dfrac{x-1}{2016}-1\right)+\left(\dfrac{x-2}{2015}-1\right)+\left(\dfrac{x-3}{2014}-1\right)+\left(\dfrac{x-4}{2013}-1\right)+\left(\dfrac{x-5}{2012}-1\right)=0\)\(\Rightarrow\dfrac{x-2017}{2016}+\dfrac{x-2017}{2015}+\dfrac{x-2017}{2014}+\dfrac{x-2017}{2013}+\dfrac{x-2017}{2012}=0\)\(\Rightarrow\left(x-2017\right)\left(\dfrac{1}{2016}+\dfrac{1}{2015}+\dfrac{1}{2014}+\dfrac{1}{2013}+\dfrac{1}{2012}\right)=0\)
\(\Rightarrow x-2017=0\Rightarrow x=2017\)
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\(A=\frac{\left(1+2+3+...+100\right)\left(\frac{1}{4}+\frac{1}{6}-\frac{1}{2}\right)\left(63.1,2-21.3,6+1\right)}{1-2+3-4+....+99-100}\)
\(=\frac{\frac{100\left(100+1\right)}{2}\left(\frac{3+2-6}{12}\right)\left[63\left(1,2-1,2\right)+1\right]}{\left(1-2\right)+\left(3-4\right)+....+\left(99-100\right)}\)
\(=\frac{5050.\left(-\frac{1}{12}\right).1}{-1+\left(-1\right)+\left(-1\right)+...+\left(-1\right)}\)
\(=\frac{2525.\left(-\frac{1}{6}\right)}{-50}=\frac{101}{12}\)
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\(S=1+5^2+5^4+...+5^{2014}\)
\(\Rightarrow25S=5^2+5^4+...+5^{2014}+5^{2016}\)
\(\Rightarrow25S-5^{2016}+1=1+5^2+5^4+...+5^{2014}\)
\(\Rightarrow25S-5^{2016}+1=S\)
\(\Rightarrow24S=5^{2016}-1\)
\(\Rightarrow S=\frac{1}{24}\left(5^{2016}-1\right)\)