Cách 2:
Ta có: \(13A=\frac{13^{16}+13}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)
\(13B=\frac{13^{17}+13}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)
Vì \(\frac{12}{13^{16}+1}>\frac{12}{13^{17}+1}\Rightarrow1+\frac{12}{13^{16}+1}>1+\frac{12}{13^{17}+1}\)
\(\Rightarrow13A>13B\)
\(\Rightarrow A>B\)
Vậy A > B
Trước khi làm bài này ; ta chứng minh công thức sau :
Nếu \(\frac{a}{b}< 1\) thì \(\frac{a}{b}< \frac{a+m}{b+m}\)
Ta có :
\(\frac{a}{b}< 1\\ \Rightarrow a< b\\ \Rightarrow an< bn\\\Rightarrow an+ab< bn+ab\\ \Rightarrow a\left(b+n\right)< b\left(a+n\right)\\ \Rightarrow\frac{a}{b}< \frac{a+n}{b+n}\left(\text{đ}pcm\right)\)
Rồi ta áp dụng công thức này vào bài nhé !!!
\(B=\frac{13^{16}+1}{13^{17}+1}< \frac{13^{16}+1+12}{13^{17}+1+12}\\ =\frac{13^{16}+13}{13^{17}+13}\\ =\frac{13\left(13^{15}+1\right)}{13\left(13^{16}+1\right)}\\ =\frac{13^{15}+1}{13^{16}+1}< A\\ \Rightarrow B< A\)
Chúc bạn học tốt !!!!!!
Ta có:
\(13A=\frac{13.\left(13^{15}+1\right)}{13^{16}+1}=\frac{13^{16}+13}{13^{16}+1}=\frac{13^{16}+1+12}{13^{16}+1}=\frac{13^{16}+1}{13^{16}+1}+\frac{12}{13^{16}+1}=1+\frac{12}{13^{16}+1}\)
\(13B=\frac{13.\left(13^{16}+1\right)}{13^{17}+1}=\frac{13^{17}+13}{13^{17}+1}=\frac{13^{17}+1+12}{13^{17}+1}=\frac{13^{17}+1}{13^{17}+1}+\frac{12}{13^{17}+1}=1+\frac{12}{13^{17}+1}\)
\(\Rightarrow\) 13A>13B \(\Rightarrow\) A>B
