E:7456 (Mirela-Petrina Timiș)
Să se arate că fracția F = 23 n + 1 − 4 n + 1 ⋅ 19 n − 7 n ⋅ 23 + 2 2 n + 2 ⋅ 3 n 41 n + 1 − 5 2 n ⋅ 41 {\displaystyle F={\frac {23^{n+1}-4^{n+1}\cdot 19^{n}-7^{n}\cdot 23+2^{2n+2}\cdot 3^{n}}{41^{n+1}-5^{2n}\cdot 41}}} se simlifică prin 16 {\displaystyle 16}
Soluție
Pentru orice F = 23 n + 1 − 4 n + 1 ⋅ 19 n − 7 n ⋅ 23 + 2 2 n + 2 ⋅ 3 n 41 n + 1 − 5 2 n ⋅ 41 {\displaystyle F={\frac {23^{n+1}-4^{n+1}\cdot 19^{n}-7^{n}\cdot 23+2^{2n+2}\cdot 3^{n}}{41^{n+1}-5^{2n}\cdot 41}}}
se simlifică prin 
