5.
7, a, b, 16
Z własności ciągu arytmetycznego:
[tex]a = \frac{7+b}{2} \ \ |\cdot2\\\\b = \frac{a+16}{2} \ \ |\cdot 2\\\\\\2a = 7+b\\2b=a+16 \\\\2a-b = 7 \ \ |\cdot2\\-a+2b=16\\\\4a-2b=14\\-a+2b = 16\\-------- \ \ + \ \ (dodajemy \ stronami)\\\\3a = 30 \ \ /:3\\\\\boxed{a = 10}\\\\b = \frac{10+16}{2} =\frac{26}{2}\\\\\boxed{b = 16}[/tex]
6.
[tex]a_1 = 18\\a_2 = 6\\q = \frac{a_2}{a_1} = \frac{6}{18} = \frac{1}{3}\\S_6 = ?\\\\S_{n} = a_1\cdot\frac{1-q^{n}}{1-q}\\\\S_6 = 18\cdot\frac{1-(\frac{1}{3})^{6}}{1-\frac{1}{3}} = 18\cdot\frac{1-\frac{1}{729}}{\frac{2}{3}}=18\cdot\frac{3}{2}\cdot(\frac{729}{729}-\frac{1}{729})=27\cdot\frac{728}{729} =\frac{728}{27}=26\frac{26}{27}\\\\\boxed{S_6 = 26\frac{26}{27}}[/tex]
