24)
[tex]a_{n}=(-1)^{n}(-n-2)^{4}\\\\a_8 = (-1)^{8}\cdot(-8-2)^{4}\\\\a_8 = 1\cdot(-10)^{4}\\\\\boxed{a_8 = 10 \ 0000}[/tex]
25)
a)
[tex]a_1 = 50\\S_{14} = 518\\n = 14\\r = ?\\\\S_{n} = \frac{2a_1+(n-1)r}{2}\cdot n\\\\S_{14} = \frac{2\cdot50+(14-1)r}{2}\cdot14\\\\518 = (100+13r)\cdot7 \ \ /:7\\\\74=100+13r\\\\13r = 74-100\\\\13r = -26 \ \ /:13\\\\\boxed{r = -2}[/tex]
b)
[tex]a_1 = -2\\a_2 = 1\\r = a_2-a_1 = 1-(-2) = 1+2 = 3\\n = 14\\S_{14} = ?\\\\S_{n} = \frac{2a_1+(n-1)r}{2}\cdot n\\\\S_{14} = \frac{2\cdot(-2)+(14-1)\cdot3}{2}\cdot14\\\\S_{14}=(-4+13\cdot3)\cdot7\\\\S_{14} = (-4+39)\cdot7\\\\S_{14} =35\cdot7\\\\\boxed{S_{14} = 245}[/tex]
c)
[tex]a_8 = a_1+7r = 5\\a_{12} = a_1+11r = 25\\a_{100} = ?\\\\a_1 + 11r = 25\\a_1+7r = 5\\------- \ \ (odejmujemy \ stronami)\\4r = 20 \ \ /:4\\\underline{r = 5}\\\\a_1 = 5-7r = 5 - 7\cdot5 = 5 - 35\\\underline{a_1 = -30}\\\\a_{n} = a_1 + (n-1)r[/tex]
[tex]a_{100} = a_1+(100-1)r\\\\a_{100} = a_1 + 99r\\\\a_{100} = -30+99\cdot5\\\\a_{100} = -30+495\\\\\boxed{a_{100} = 465}[/tex]
d)
[tex]a_1 = 3x+1\\a_2 = 2x-4\\a_3 = 5x+3\\\\a_2 = \frac{a_1+a_3}{2}\\\\2x-4 = \frac{3x+1+5x+3}{2}\\\\2x-4 = \frac{8x+4}{2}\\\\2x-4 = 4x+2\\\\2x-4x = 2+4\\\\-2x = 6 \ \ /:(-2)\\\\\boxed{x = -3}[/tex]
