zad. 1
[tex] a_{5} = \frac{4}{9} \\ \frac{4}{9} = a_{1} \times ( - \frac{1}{3} {)}^{5 - 1} \\ \frac{4}{9} = a_{1} \times ( - \frac{1}{3} {)}^{4} \\ \frac{4}{9} = a_{1} \times \frac{1}{81} \\a_{1} = 36[/tex]
zad. 2
[tex] a_{1} =4 \\ q = \frac{1}{2} \\ S_{8} =4 \times \frac{1 - ( \frac{1}{2} {)}^{8} }{1 - \frac{1}{2} } = 4 \times \frac{1 - \frac{1}{256} }{ \frac{1}{2} } = 4 \times \frac{ \frac{255}{256} }{ \frac{1}{2} } = 4 \times \frac{255}{128} = \frac{255}{32} [/tex]
zad. 3
[tex]a_{3} = 2 \\ a_{6} = 16 \\ \\ 2 = a_{1} \times {q}^{2} \: \: \: \: | \div {q}^{2} \\ 16 = a_{1} \times {q}^{5} \\ \\a_{1} = \frac{2}{ {q}^{2} } \\ 16 = \frac{2}{ {q}^{2} } \times {q}^{5} \\ \\ a_{1} = \frac{2}{ {q}^{2} } \\ 16 = 2 {q}^{3} \: \: \: | \div 2 \\ \\ a_{1} = \frac{2}{ {q}^{2} } \\ {q}^{3} = 8 \\ \\ a_{1} = \frac{2}{ {q}^{2} } \\ q = 2 \\ \\ a_{1} = \frac{2}{ {2}^{2} } \\ q = 2 \\ \\ a_{1} = \frac{1}{2} \\ q = 2[/tex]
[tex]S_{10} = \frac{1}{2} \times \frac{1 - {2}^{10} }{1 - 2} = \frac{1}{2} \times \frac{1 - 1024}{ - 1} = \frac{1}{2} \times 1023 = \frac{1023}{2} = 511 \frac{1}{2} [/tex]
zad. 4
