Explicație pas cu pas:
10.
[tex]a = {4}^{n + 1} \cdot {3}^{n} \cdot 6 + {4}^{n + 1} \cdot {3}^{n + 1} \cdot 3 + {2}^{2n + 1} \cdot {3}^{n} \cdot 2 \cdot 2 = \\ = {4}^{n + 1} \cdot {3}^{n} \cdot 6 + {4}^{n + 1} \cdot {3}^{n} \cdot 3 \cdot 3 + {2}^{2(n +1) } \cdot {3}^{n} \cdot 2 \\ = {4}^{n + 1} \cdot {3}^{n} \cdot 6 + {4}^{n + 1} \cdot {3}^{n} \cdot 9 + {4}^{n +1} \cdot {3}^{n} \cdot 2 \\ = {4}^{n + 1} \cdot {3}^{n}\cdot (6 + 9 + 2) = \red {\bf 17} \cdot {4}^{n + 1} \cdot {3}^{n}[/tex]
11.
[tex]\left( \frac{5}{9} \div \frac{16 - 1}{9} - \frac{16 - 1}{90} \div \frac{1}{2}\right) \cdot \left( \frac{3}{16} \cdot \frac{16}{ {3}^{4} } - \frac{1}{ {3}^{3} }\right) = \\ [/tex]
[tex]= \left( \frac{5}{9} \cdot \frac{9}{15} - \frac{15}{90} \cdot 2\right) \cdot \left(\frac{1}{ {3}^{3} } - \frac{1}{ {3}^{3} }\right) \\ [/tex]
[tex]= \left( \frac{1}{3} - \frac{1}{3} \right) \cdot \left(\frac{1}{ {3}^{3} } - \frac{1}{ {3}^{3} }\right) = 0\cdot0 = \red {\bf 0} \\ [/tex]
12.
[tex]n = 12a + 5 \iff n - 5 = {2}^{2} \cdot 3 \cdot a[/tex]
[tex]n = 9b + 5 \iff n - 5 = {3}^{2} \cdot b[/tex]
[tex]n = 8c + 5\iff n - 5 = {2}^{3} \cdot c[/tex]
[tex]c.m.m.m.c. [12,9,8] = {2}^{3} \times {3}^{2} = 72[/tex]
[tex]n - 5 = 72 \iff n = 72 + 5 \\ \implies \red{\bf n = 77}[/tex]
13.
a)
[tex]x + 15 + 2x - 6 + 3x - 15 = 360 \\ 6x = 366 \iff x = 61[/tex]
∢AOB = 61 + 15 = 76°
∢BOC = 2•61 - 6 = 116°
∢AOC = 3•61 - 15 = 168°
b) desen
c) ∢MOC = 180° - ∢BOC = 180° - 116° = 64°
