Question

aratati ca pentru orice n apartine N, numarul E = 2 la 2n+3 *25 la n - 4 la n *5 la 2n este divizibil cu sapte​

Answer 1

Sper că te-am ajutat!

Answer 2

$E = 2^{2n+3}\cdot 25^{n}-4^{n}\cdot 5^{2n} \\ \\ E= 2^{2n+3}+5^{2n}-2^{2n}\cdot 5^{2n}\\ \\ E = 5^{2n}\cdot (2^{2n+3}-2^{2n}) \\ \\ E = 5^{2n}\cdot \Big[2^{2n}\cdot (2^3-1)\Big]\\ \\ E = 5^{2n}\cdot 2^{2n}\cdot (2^3-1) \\ \\ E = 10^{2n}\cdot 7 \\ \\ \Rightarrow E \,\,\vdots\,\,7$

Sau altfel:

$\\ \\E = 2^{2n+3}\cdot (21+4)^{n}-4^n\cdot (7-2)^{2n} \\ \\ E = 2^{2n+3}\cdot (M_7+4^n) - 4^n\cdot \Big(M_7+(-2)^{2n}\Big)\\ \\ E = M_{7}+2^{4n+3}-M_7-2^{4n} \\ \\ E = M_7+2^{4n}\cdot(2^{3}-1)\\ \\ E = M_{7}+2^{4n}\cdot 7 \\ \\ E = M_{7}$