Question

Không quy đồng , hẫy so sánh

\(\frac{5^{12}+1}{5^{13}+1}\) và \(\frac{5^{11}+1}{5^{12}+1}\)

Answer 1

công thức \(\frac{a}{b}< \frac{a+m}{b+m}\)

nên ta có :  \(\frac{5^{12}+1}{5^{13}+1}< \frac{5^{12}+1+4}{5^{13}+1+4}\)\(=\frac{5^{12}+5}{5^{13}+5}=\frac{5.\left(5^{11}+1\right)}{5.\left(5^{12}+1\right)}=\frac{5^{11}+1}{5^{12}+1}\)

=> \(\frac{5^{12}+1}{5^{13}+1}< \frac{5^{11}+1}{5^{12}+1}\)

Answer 2

đặt A và B = 2 cái kia rồi nhân nó với 5 là đc

Answer 3

Áp dụng a/b < 1 => a/b < a+m/b+m (a,b,m thuộc N*)

Do 5¹² + 1/5¹³ + 1 < 5¹² + 1 + 4/5¹³ + 1 + 4

                            < 5¹² + 5/5¹³ + 5

                           < 5.(5¹¹ + 1)/5.(5¹² + 1)

                          < 5¹¹ + 1/5¹² + 1

Vậy 5¹² + 1/5¹³ + 1 < 5¹¹ + 1/5¹² + 1

Ủng hộ mk nha ^_^
 

Answer 4

Đặt \(A=\frac{5^{12}+1}{5^{13}+1}\)và \(B=\frac{5^{11}+1}{5^{12}+1}\)

\(5A=\frac{5\left(5^{12}+1\right)}{5^{13}+1}=\frac{5^{13}+5}{5^{13}+1}=\frac{5^{13}+1+4}{5^{13}+1}=\frac{5^{13}+1}{5^{13}+1}+\frac{4}{5^{13}+1}=1+\frac{4}{5^{13}+1}\)

\(5B=\frac{5\left(5^{11}+1\right)}{5^{12}+1}=\frac{5^{12}+5}{5^{12}+1}=\frac{5^{12}+1+4}{5^{12}+1}=\frac{5^{12}+1}{5^{12}+1}+\frac{4}{5^{12}+1}=1+\frac{4}{5^{12}+1}\)

Vì 5¹³+1>5¹²+1 suy ra \(\frac{4}{5^{13}+1}< \frac{4}{5^{12}+1}\)

\(\Rightarrow1+\frac{4}{5^{13}+1}< 1+\frac{4}{5^{12}+1}\)

\(\Rightarrow5A< 5B\)

\(\Rightarrow A< B\)