Gọi \(\frac{13^{15}+1}{13^{16}+1}\)là S, \(\frac{13^{16}+1}{13^{17}+1}\)là X
\(13\cdot S=13\cdot\frac{13^{15+1}}{13^{16}+1}=\frac{13.\left(13^{15}+1\right)}{13^{16}+1}=\frac{13^{16}+13}{13^{16}+1}\)\(=\frac{13^{16}+1+12}{13^{16}+1}=\frac{13^{16}+1}{13^{16}+1}+\frac{12}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)
\(13\cdot X=13.\frac{13^{16}+1}{13^{17}+1}=\frac{13\cdot\left(13^{16}+1\right)}{13^{17}+1}=\frac{13^{17}+13}{13^{17}+1}\)\(=\frac{13^{17}+1+12}{13^{17}+1}=\frac{13^{17}+1}{13^{17}+1}+\frac{12}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)
Do \(1+\frac{12}{13^{16}+1}>1+\frac{12}{13^{17}+1}\)\(\rightarrow13\cdot S>13\cdot X\)\(\rightarrow S>X\)