[tex]a_1 = 6\\q = \frac{3}{2}\\S_{n} = 193\frac{1}{32} = \frac{6177}{32}\\n = ?\\\\S_{n} = a_1\cdot\frac{1-q^{n}}{1-q}\\\\\frac{6177}{32} = 6\cdot\frac{1-(\frac{3}{2})^{n}}{1-\frac{3}{2}}\\\\\frac{6177}{32} = 6\cdot\frac{1-\frac{3}{2})^{n}}{\frac{-1}{2}}\\\\\frac{6177}{32} = -12[1-(\frac{3}{2})^{n}] \ \ |\cdot32\\\\-384[1-(\frac{3}{2})^{n}] = 6177\\\\-384 + 384\cdot(\frac{3}{2})^{n} = 6177\\\\384\cdot(\frac{3}{2})^{n} =6177+384\\\\384\cdot(\frac{3}{2})^{n} = 6561 \ \ /:3[/tex]
[tex]128\cdot(\frac{3}{2})^{n}=2187 \ \ /:128\\\\(\frac{3}{2})^{n} = \frac{2187}{128}\\\\(\frac{3}{2})^{n} = \frac{3^{7}}{2^{7}}\\\\(\frac{3}{2})^{n} = (\frac{3}{2})^{7}\\\\\boxed{n = 7}[/tex]
Odp. Należy dodać 7 początkowych wyrazów tego ciągu.
