(aₙ) - ciąg geometryczny
aₙ = a₁ · qⁿ⁻¹, gdzie q to iloraz ciągu geometrycznego
[tex]a_3 = 16 \ i \ a_3 = a_1 \cdot q^2 \\ a_1 \cdot q^2 = 16 \ \ \ |:q^2 \\ a_1 = \frac{16}{q^2} \\\\ a_6=2 \ i \ a_6 = a_1 \cdot q^5 \\ a_1 \cdot q^5 = 2 \ \ \ |:q^5 \\ a_1 = \frac{2}{q^5} \\\\ \frac{2}{q^5} = \frac{16}{q^2} \\ 16q^5 = 2q^2 \ \ \ |:16q^2 \\q^3 = \frac{2}{16} \\q^3= \frac{1}{8} \\ q = \sqrt{\frac{1}{8}} \\ \underline{q=\frac{1}{2}} \\\\ a_1 = \frac{16}{q^2} \\ a_1 = \frac{16}{(\frac{1}{2})^2} \\ a_1 = \frac{16}{\frac{1}{4}} \\ a_1 = 16 : \frac{1}{4}\\ a_1 = 16 \cdot 4 \\ \underline{a_1 = 64}[/tex]
Suma 10-tu pierwszych wyrazów ciągu
[tex]S_{10}= a_1 \cdot \dfrac{1-q^{10}}{1 - q} =64 \cdot \dfrac{1-(\frac{1}{2})^{10}}{1 - \frac{1}{2}} = 64 \cdot \dfrac{1-\frac{1}{1024}}{\frac{1}{2}} = 64 \cdot(\frac{1024}{1024}-\frac{1}{1024}) :\frac{1}{2} = \\\\ = \not{64}^1 \cdot \frac{1023}{\not{1024}_{16}} \cdot 2 = \frac{2046}{16} = 127,875[/tex]
Wzór ogólny ciągu
[tex]a_n = a_1 \cdot q^{n - 1} = 64 \cdot (\frac{1}{2})^{n - 1}= 64 \cdot (\frac{1}{2})^n : (\frac{1}{2})^1 = 64 \cdot \frac{1}{2^n}: \frac{1}{2}= \frac{64}{2^n} \cdot 2= \frac{128}{2^n} = \\ = \frac{2^7}{2^n}= 2^{7-n} \\ a_n = 2^{7-n}[/tex]
