Question

Arătați că, pentru orice număr natural n numarul natural N=5*7 la n-3*7 la n+1 +7 la n+2 este divizibil cu 11

Answer 1

Răspuns:

Explicație pas cu pas:

Answer 2

 

$\displaystyle\bf\\N=5\times7^n-3\times7^{n+1}+7^{n+2}\\\\N=5\times7^n-3\times7^n\times7^1+7^n\times 7^2\\\\N=5\times7^n-3\times7^n\times7+7^n\times 49\\\\N=5\times7^n-3\times7\times 7^n+49\times7^n\\\\N=5\times7^n-21 \times 7^n+49\times7^n\\\\N=7^n\Big(\underbrace{5-21+49}_{=~33}\Big)\\\\N=33\times7^n\\\\N=11\times3\times7^n\\\\\implies~~\boxed{\bf N~\vdots~11,~oricare~ar~fi~n~\in~R}$