Question

Aratati ca fractia: b) 2^n+3+2^n+2+2^n+1+2^n/3^n+2+2x3^n+1+3^n se simplifica prin 6, pentru orice n aprtine N *am ajuns pana la 2^nx15/3^nx16, de acolo nu mai stiu, pls si explicati ce faceti**.

Answer 1

Răspuns:

divizibil cu 6

Explicație pas cu pas:

$\frac{ {2}^{n + 3} + {2}^{n + 2} + {2}^{n + 1} + {2}^{n}}{ {3}^{n + 2} + 2 \cdot {3}^{n + 1} + {3}^{n} } = \frac{ {2}^{n} \cdot ( {2}^{3} + {2}^{2} + {2}^{1} + {2}^{0})}{ {3}^{n} \cdot ( {3}^{2} + 2 \cdot {3}^{1} + {3}^{0})} = \\ = \frac{ {2}^{n} \cdot (8 + 4 + 2 + 1)}{ {3}^{n} \cdot (9 + 6 + 1)} = \frac{15 \cdot 2 \cdot {2}^{n - 1} }{16 \cdot 3 \cdot {3}^{n - 1} } = \frac{6 \cdot 5 \cdot {2}^{n - 1} }{6 \cdot 8 \cdot {3}^{n - 1} } \ \ \red{\bf \vdots \ \ 6}$

q.e.d.