## 2016年12月20日火曜日

### 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)

1. $\begin{array}{l}a{r}^{2}=4\\ a{r}^{4}=36\\ {r}^{2}=9\\ r=±3\\ a=\frac{4}{9}\\ {a}_{n}=\frac{4}{9}·{\left(±3\right)}^{n-1}=4·{\left(±3\right)}^{r-3}\end{array}$

2. $\begin{array}{l}a{r}^{2}=9\\ a{r}^{5}=-\frac{8}{3}\\ {r}^{3}=-\frac{8}{3}·\frac{1}{9}=-{\left(\frac{2}{3}\right)}^{3}\\ r=-\frac{2}{3}\\ a\frac{4}{9}=9\\ a=\frac{81}{4}\\ {a}_{n}=\frac{81}{4}·{\left(-\frac{2}{3}\right)}^{n-1}=9{\left(-\frac{2}{3}\right)}^{n-3}\end{array}$

1. $\begin{array}{l}\frac{b}{a}=\frac{c}{b}\\ {b}^{2}=ac\end{array}$

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

$\begin{array}{l}{a}_{n}=a{r}^{n-1}\\ {b}_{n}=b{s}^{n-1}\\ {a}_{n}{b}_{n}=ab{\left(rs\right)}^{n-1}\end{array}$

1. $\begin{array}{l}{a}_{n}=A+dn\\ \\ {10}^{{a}^{n}}={10}^{A+dn}\\ ={10}^{A}·{\left({10}^{d}\right)}^{n}\end{array}$

2. $\begin{array}{l}{b}_{n}=a{r}^{n-1}\\ {\mathrm{log}}_{10}{b}_{n}={\mathrm{log}}_{10}a{r}^{n-1}\\ ={\mathrm{log}}_{10}a+\left(n-1\right){\mathrm{log}}_{10}r\\ =\left({\mathrm{log}}_{10}a-{\mathrm{log}}_{10}r\right)+n{\mathrm{log}}_{10}r\end{array}$