Question

Arătaţi că (4^n•3^n+1 +12^n+1 -6^n•2^n+1) se imparte exact la 13​

Answer 1

Explicație pas cu pas:

${4}^{n} \cdot {3}^{n + 1} + {12}^{n + 1} - {6}^{n} \cdot {2}^{n + 1} =$

$= {4}^{n} \cdot {3}^{n} \cdot 3 + 12 \cdot {12}^{n} - {6}^{n} \cdot {2}^{n} \cdot 2$

$= 3 \cdot {(4 \cdot 3)}^{n} + 12 \cdot {12}^{n} - 2 \cdot {(6 \cdot 2)}^{n}$

$= 3 \cdot {12}^{n} + 12 \cdot {12}^{n} - 2 \cdot {12}^{n}$

$= {12}^{n} \cdot (3 + 12 - 2)$

$= \bf 13 \cdot {12}^{n} \ \ \vdots \ 13$

q.e.d.