Question

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So sánh\(\frac{7^{58}+2}{7^{57}+2}và\frac{7^{57}+2017}{7^{56}+2017}\)

Answer 1

[Nguyen Van Huong]

Ta có : ''Phần hơn'' của \(\frac{7^{58}+2}{7^{57}+2}\) là :

             \(\frac{7^{58}+2}{^{ }7^{57}+2}\) \(-\) 1 = \(\frac{7^{57}.6}{7^{57}+2}\)

             ''Phần hơn'' của \(\frac{5^{57}+2017}{5^{56}+2017}\) với 1 là :

             \(\frac{7^{57}+2017}{7^{56}+2017}\) \(-\) 1 = \(\frac{7^{56}.6}{7^{56}+2017}\)

           Ta có :\(\frac{7^{56}.6}{7^{56}+2017}\) = \(\frac{7^{56}.7.6}{\left(7^{56}+2017\right)7}\) = \(\frac{7^{57}.6}{7^{57}+14119}\)

         Ta thấy \(\frac{7^{57}.6}{7^{57}+2}\)> \(\frac{7^{57}.6}{7^{57}+14119}\)

         Suy ra \(\frac{7^{57}.6}{7^{57}+2}\) > \(\frac{7^{56}.6}{7^{56}+2017}\)

         Do đó \(\frac{7^{58}+2}{7^{57}+2}\) > \(\frac{7^{57}+2017}{7^{56}+2017}\)