Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 181

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\[1) - 16;\ - 4;\ - 1;\ldots\ \ \ \ n = 5\]

\[b_{1} = - 16;\ \ b_{2} = - 4;\]

\[q = \frac{b_{2}}{b_{1}} = \frac{1}{4}.\]

\[S_{5} = \frac{b_{1} \cdot \left( 1 - q^{5} \right)}{1 - q} =\]

\[= \frac{- 16 \cdot \left( 1 - \left( \frac{1}{4} \right)^{5} \right)}{1 - \frac{1}{4}} =\]

\[= \frac{- 16 \cdot \left( 1 - \frac{1}{1024} \right)}{\frac{3}{4}} =\]

\[= \frac{- 16 \cdot 4 \cdot \frac{1023}{1024}}{3} = \frac{- 64 \cdot 1023}{3 \cdot 1024} =\]

\[= - \frac{341}{16} = - 21\frac{5}{16}.\]

\[2)\ \frac{1}{16};\ - \frac{1}{8};\ \frac{1}{4};\ldots\ \ \ n = 6\]

\[b_{1} = \frac{1}{16};\ \ \ b_{2} = - \frac{1}{8};\]

\[q = - \frac{1}{8}\ :\frac{1}{16} = - \frac{16}{8} = - 2.\]

\[S_{6} = \frac{b_{1} \cdot \left( 1 - q^{6} \right)}{1 - q} =\]

\[= \frac{\frac{1}{16} \cdot \left( 1 - ( - 2)^{6} \right)}{1 + 2} = \frac{1 - 64}{16 \cdot 3} =\]

\[= - \frac{63}{48} = - \frac{21}{16} = - 1\frac{5}{16}.\]