1.
a)
[tex]a_1 =1024 = 2^{10}\\q = \frac{1}{2}\\a_6 = ?\\\\a_{n} = a_1\cdot q^{n-1}\\\\a_6 = 2^{10}\cdot \frac{1}{2}^{5}=2^{10}\cdot2^{-5}\\\\\boxed{a_6 = 2^{5} = 32}[/tex]
b)
[tex]S_{10} = ?\\\\S_{n} = a_1\cdot\frac{1-q^{n}}{1-q}\\\\S_{10} =1024\cdot\frac{1-(\frac{1}{2})^{6}}{1-\frac{1}{2}} = 1024\cdot\frac{1-\frac{1}{64}}{\frac{1}{2}}=1024\cdot\frac{\frac{64}{64}-\frac{1}{64}}{\frac{1}{2}}=1024\cdot\frac{63\cdot2}{64}=1024\cdot\frac{126}{64}\\\\\boxed{S_{10} = 2016}[/tex]
2.
[tex]cos3\alpha = 4cos3\alpha - 3cos3\alpha\\\\P = 4cos3\alpha - 3cos3\alpha = cos3\alpha\\\\L = cos3\alpha}\\\\L = P[/tex]
