Question

So sánh: \(\frac{3^{17}+1}{3^{20}+1}\)và \(\frac{3^{20}+1}{3^{23}+1}\)

Answer 1

Coi:

\(A=\frac{3^{17}+1}{3^{20}+1}\)và\(B=\frac{3^{20}+1}{3^{23}+1}\)

Có:\(A=\frac{3^{17}+1}{3^{20}+1}=\left(\frac{3^{17}+1}{3^{20}+1}\right).\frac{3^3}{3^3}=\frac{3^{20}+27}{3^{23}+27}\)

Ta lại có:

\(1-A=1-\frac{3^{20}+27}{3^{23}+27}=\frac{3^{23}+27}{3^{23}+27}-\frac{3^{20}+27}{3^{23}+27}=\frac{3^{23}-3^{20}}{3^{23}+27}\)

\(1-B=1-\frac{3^{20}+1}{3^{23}+1}=\frac{3^{23}+1}{3^{23}+1}-\frac{3^{20}+1}{3^{23}+1}=\frac{3^{23}-3^{20}}{3^{23}+1}\)

Vì\(\frac{3^{23}-3^{20}}{3^{23}+27}\frac{3^{20}+1}{3^{23}+1}\)