[tex]a_6 = -\frac{1}{25}=a_1\cdot q^{5}\\a_3 = 5 = a_1\cdot q^{2}\\--------- \ \ dzielimy \ stronami\\\\q^{3} = \frac{-\frac{1}{25}}{5}\\\\q^{3} = -\frac{1}{125}\\\\q = \sqrt[3]{-\frac{1}{125}}=\sqrt[3]{(-\frac{1}{5})^{3}}}\\\\\boxed{q = -\frac{1}{5}}\\\\a_1\cdot q^{2} = a_3\\\\a_1 \cdot(-\frac{1}{5})^{2} = 5\\\\\frac{1}{25}a_1 = 5 \ \ /\cdot25\\\\\boxed{a_1 = 125}\\\\\\a_{n} = a_1\cdot q^{n-1}\\\\a_2 = a_1\cdot q=125\cdot(-\frac{1}{5}) = -25\\\\a_3 = 5\\\\a_4 = a_3\cdot q =5\cdot(-\frac{1}{5}) = -1[/tex]
[tex]a_5 = a_4\cdot q = -1\cdot(-\frac{1}{5})=\frac{1}{5}[/tex]
[tex]S_{5} = a_1+a_2+a_3+a_4+a_5\\\\S_{5} = 125-25+5-1+\frac{1}{5} = 104+\frac{1}{5}\\\\\boxed{S_5 = 104\frac{1}{5}}[/tex]
