\(A=\frac{100^{2016}+1}{100^{2015}-1}\)
\(\frac{1}{100}.A=\frac{100^{2016}+1}{100\left(100^{2015}-1\right)}\)
\(=\frac{100^{2016}+1}{100^{2016}-100}\)
\(=\frac{\left(100^{2016}-100\right)+101}{100^{2016}-100}\)
\(=\frac{100^{2016}-100}{100^{2016}-100}\)\(+\frac{101}{100^{2016}-100}\)
\(=1+\frac{101}{100^{2016}-100}\)
\(B=\frac{100^{2015}+1}{100^{2014}-1}\)
\(\frac{1}{100}.B=\frac{100^{2015}+1}{100\left(100^{2014}-1\right)}\)
\(=\frac{100^{2015}+1}{100^{2015}-100}\)
\(=\frac{\left(100^{2015}-100\right)+101}{100^{2015}-100}\)
\(=\frac{100^{2015}-100}{100^{2015}-100}\)\(+\frac{101}{100^{2015}-100}\)
\(=1+\frac{101}{100^{2015}-100}\)
\(\hept{\begin{cases}Vì101>0\\100^{2016}-100>100^{2015}-100>0\end{cases}}\)
\(\Rightarrow\frac{101}{100^{2016}-100}< \frac{101}{100^{2015}-100}\)
\(\Rightarrow1+\frac{101}{100^{2016}-100}< 1+\frac{101}{100^{2015}-100}\)
\(\Rightarrow\frac{1}{100}.A< \frac{1}{100}.B\)
\(\Rightarrow A< B\left(vì\frac{1}{100}>0\right)\)
Vậy A<B
