We must compare \(\left(10^{2016}+1\right)^2\)and \(\left(10^{2015}+1\right)\left(10^{2017}+1\right)\).

We have : \(\left(10^{2015}+1\right)\left(10^{2017}+1\right)-\left(10^{2016}+1\right)^2\)

\(=10^{4032}+10^{2015}+10^{2017}+1-10^{4032}-2.10^{2016}-1\)

\(=10^{2015}-10^{2016}+10^{2017}-10^{2016}\)

\(=10^{2015}\left(1-10+10^2-10\right)>0\)

\(\Rightarrow\left(10^{2015}+1\right)\left(10^{2017}+1\right)>\left(10^{2016}+1\right)^2\)

\(\Rightarrow\dfrac{10^{2016}+1}{10^{2017}+1}< \dfrac{10^{2015}+1}{10^{2016}+1}\)