## 2017年1月18日水曜日

### 数学 - 「離散的」な世界 - 数列 – 数学的帰納法と数列 - 帰納的定義(一般項)

1. $\begin{array}{l}{b}_{n}={a}_{n+1}-{a}_{n}\\ ={a}_{n}+2n-{a}_{n}\\ =2n\\ \\ n>1\\ {a}_{n}={a}_{1}+\sum _{i=1}^{n-1}{b}_{i}\\ =1+2·\frac{\left(n-1\right)n}{2}\\ ={n}^{2}-n+1\\ {1}^{2}-1+1=1\\ \\ {a}_{n}={n}^{2}-n+1\end{array}$

2. $\begin{array}{l}{b}_{n}={a}_{n+1}-{a}_{n}\\ ={a}_{n}+{3}^{n}-{a}_{n}\\ ={3}^{n}\\ \\ n>1\\ {a}_{n}={a}_{1}+\sum _{i=1}^{n-1}{b}_{n}\\ =2+\frac{3\left({3}^{n-1}-1\right)}{3-1}\\ =\frac{{3}^{n}}{2}+\frac{1}{2}\\ \\ \frac{3}{2}+\frac{1}{2}=2\\ {a}_{n}=\frac{1}{2}\left({3}^{n}+1\right)\end{array}$

3. $\begin{array}{l}x=3x-2\\ x=1\\ \\ {a}_{n+1}-1=3\left({a}_{n}-1\right)\\ \\ {a}_{n}-1=\left(2-1\right)·{3}^{n-1}\\ {a}_{n}={3}^{n-1}+1\end{array}$

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

5. $\begin{array}{l}2x=x-6\\ x=-6\\ 2\left({a}_{n+1}+6\right)={a}_{n}+6\\ {a}_{n+1}+6=\frac{1}{2}\left({a}_{n}+6\right)\\ \\ {a}_{n}+6=\left(2+6\right)·\frac{1}{{2}^{n-1}}\\ {a}_{n}=\frac{1}{{2}^{n-4}}-6\end{array}$

6. $\begin{array}{l}1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5}\\ \\ {a}_{n+1}=\frac{n}{n+1}·\frac{1}{n}=\frac{1}{n+1}\\ \\ {a}_{n}=\frac{1}{n}\end{array}$

7. $\begin{array}{l}2,\frac{2}{3},\frac{2}{5},\frac{2}{7},\frac{2}{9}\\ \\ {a}_{n+1}=\frac{\frac{2}{2n-1}}{\frac{2}{2n-1}+1}\\ =\frac{2}{2n-1}·\frac{2n-1}{2n+1`}\\ =\frac{2}{2\left(n+1\right)-1}\\ \\ {a}_{n}=\frac{2}{2n-1}\end{array}$

8. $\begin{array}{l}3,3,6,12,24\\ \\ n>1\\ {a}_{n+1}={a}_{1}+{a}_{2}+···+{a}_{n-1}+{a}_{n}\\ =2{a}_{n}\\ {a}_{n}=3·{2}^{n-2}\\ \\ {a}_{1}=3,{a}_{n}=3·{2}^{n-2}\left(n>1\right)\end{array}$

$\begin{array}{l}1,\frac{1}{3},\frac{1}{7},\frac{1}{15},\frac{1}{31}\\ \\ {a}_{n+1}=\frac{\frac{1}{{2}^{n}-1}}{\frac{1}{{2}^{n}-1}+2}\\ =\frac{1}{{2}^{n}-1}·\frac{{2}^{n}-1}{1+{2}^{n+1}-2}\\ =\frac{1}{{2}^{n+1}-1}\\ \\ {a}_{n}=\frac{1}{{2}^{n}-1}\end{array}$