Question

Ex 20 va rog rapid sa ma ajutați!

Answer 1

a)

$2 ^{n + 2} \times {3}^{n + 1 } + {2}^{n + 3} \times 3 ^{n} \\ 2 ^{n} \times {2}^{2} \times {3}^{n} \times 3^{1} + {2}^{n} \times {2}^{3} \times {3}^{n} \\ {2}^{n} \times {3}^{n} \times {2}^{2} \times 3 + {2}^{n} \times {3}^{n} \times 2 ^{3}$

observam ca putem aplica metoda factorului comun in cazul nostru 2^n×3^n

${2}^{n} \times {3}^{n} ( {2}^{2} \times 3 + {2}^{3} \times 1) \\ {2}^{n} \times {3}^{n} (4 \times 3 + 8) \\ {2}^{n} \times {3}^{n} \times 20$

20 divizibil cu 4=> ca tot numarul se divide cu 4

b)

${3}^{n + 1} \times {7}^{n + 1} - {3}^{n + 2} \times {7}^{n} \\ {3}^{n} \times {7}^{n} \times 3 \times 7 - {3}^{n} \times {7}^{n} \times {3}^{2} \\ {3}^{n} \times {7}^{n}(3 \times 7 - {3}^{2} ) \\ {3}^{n} \times {7}^{n} \times 4 \times 3$

deci numarul dat este multiplu de 4 deoarece se poate scrie ca X×4

cu draaag