数学 - 「離散的」な世界 - 数列 – 数列とその和 - 等比数列とその一般項(初項、公比、一般項)

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数学読本〈3〉平面上のベクトル/複素数と数列とその和/空間図形/2次曲線/数列(松坂和夫(著)、岩波書店)の第13章(「離散的」な世界 - 数列)、13.1(数列とその和)、等比数列とその一般項、問13、14、15、16.を取り組んでみる。

問13.

$\begin{array}{l} a r^{2} = 4 \\ a r^{4} = 36 \\ r^{2} = 9 \\ r = \pm 3 \\ a = \frac{4}{9} \\ a_{n} = \frac{4}{9} \cdot {(\pm 3)}^{n - 1} = 4 \cdot {(\pm 3)}^{r - 3} \end{array}$
$\begin{array}{l} a r^{2} = 9 \\ a r^{5} = - \frac{8}{3} \\ r^{3} = - \frac{8}{3} \cdot \frac{1}{9} = - {(\frac{2}{3})}^{3} \\ r = - \frac{2}{3} \\ a \frac{4}{9} = 9 \\ a = \frac{81}{4} \\ a_{n} = \frac{81}{4} \cdot {(- \frac{2}{3})}^{n - 1} = 9 {(- \frac{2}{3})}^{n - 3} \end{array}$

問14.

$\begin{array}{l} \frac{b}{a} = \frac{c}{b} \\ b^{2} = a c \end{array}$
$\begin{array}{l} a + b + c = 13 \\ a b c = 27 \\ b^{2} = a c \\ b^{3} = 27 \\ b = 3 \\ a c = 9 \\ a + 3 + c = 13 \\ c = 10 - a \\ a (10 - a) = 9 \\ a^{2} - 10 a + 9 = 0 \\ (a - 1) (a - 9) = 0 \\ a = 1, 9 \\ a = 1, b = 3, c = 9 \\ a = 9, b = 3, c = 1 \end{array}$

問15.

\begin{array}{l} a_{n} = a r^{n - 1} \\ b_{n} = b s^{n - 1} \\ a_{n} b_{n} = a b {(r s)}^{n - 1} \end{array}

問16.

$\begin{array}{l} a_{n} = A + d n \\ 10^{a^{n}} = 10^{A + d n} \\ = 10^{A} \cdot {(10^{d})}^{n} \end{array}$
$\begin{array}{l} b_{n} = a r^{n - 1} \\ \log_{10} b_{n} = \log_{10} a r^{n - 1} \\ = \log_{10} a + (n - 1) \log_{10} r \\ = (\log_{10} a - \log_{10} r) + n \log_{10} r \end{array}$

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