Odpowiedź:
[tex]\huge\boxed{q = \frac{3}{2}=1\frac{1}{2}}\\\\\\\huge\boxed{a_1 = \frac{64}{81}}\\\\\\\huge\boxed{a_9 = 20\frac{1}{4}}[/tex]
Szczegółowe wyjaśnienie:
[tex]a_1 = 4\\a_{8} = \frac{27}{2}\\q = ?\\a_9 = ?[/tex]
Korzystamy ze wzoru na n-ty wyraz ciągu geometrycznego:
[tex]a_{n} = a_1\cdot q^{n-1}\\\\a_5 = a_1 \cdot q^{4} = 4\\a_{8} = a_1\cdot q^{7} = \frac{27}{2}\\\\a_1 \cdot q^{7} =\frac{27}{2}\\a_1\cdot q^{4} = 4\\------ \ \ (dzielimy \ stronami)\\\\\frac{q^{7}}{q^{4}} = \frac{\frac{27}{2}}{4}\\\\q^{7-4} = \frac{27}{8}\\\\q^{3} = (\frac{3}{2})^{3}\\\\\boxed{q = \frac{3}{2}}[/tex]
[tex]a_5 = a_1\cdot q^{4} = 4\\\\a_1\cdot(\frac{3}{2})^{4} = 4\\\\a_1 \cdot\frac{3^{4}}{2^{4}} = 2^{2} \ \ \ |\cdot\frac{2^{4}}{3^{4}}\\\\a_1 = \frac{2^{2}\cdot 2^{4}}{3^{4}} = \frac{2^{6}}{3^{4}}\\\\\boxed{a_1 = \frac{64}{81}}[/tex]
[tex]a_9 = a_8\cdot q\\\\a_9 = \frac{27}{2}\cdot\frac{3}{2}=\frac{81}{4}\\\\\boxed{a_9 = 20\frac{1}{4}}[/tex]
