# 高中數學/不等式與數列/數列與通項公式的概念

## 基礎知識

### 數列的定義與基本概念

${\displaystyle a_{1},a_{2},a_{3},\cdots ,a_{n},\cdots }$

A.23；B.29；C.32；D.38

A.${\displaystyle a_{n}=n^{2}-n+1}$； B.${\displaystyle a_{n}={\frac {n(n-1)}{2}}}$； C.${\displaystyle a_{n}={\frac {n(n+1)}{2}}}$； D.${\displaystyle a_{n}={\frac {n(n+2)}{2}}}$

A.${\displaystyle {\frac {1}{2}}}$； B.${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}}$； C.${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}}$； D.${\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}}$

A.${\displaystyle {\frac {1}{2n+1}}}$； B.${\displaystyle {\frac {1}{2n+2}}}$； C.${\displaystyle {\frac {1}{2n+1}}+{\frac {1}{2n+2}}}$； D.${\displaystyle {\frac {1}{2n+1}}+{\frac {1}{2n+2}}}$

### 常見性質與分類

• 常數列
• 單調數列
• 擺動數列
• 周期數列

A.遞增數列；B.遞減數列；C.擺動數列；D.周期數列

## 常用結論與常見模型

### 根據規律猜想通項公式

A.${\displaystyle {\frac {16}{17}}}$； B.${\displaystyle {\frac {18}{19}}}$； C.${\displaystyle {\frac {20}{21}}}$； D.${\displaystyle {\frac {22}{23}}}$

(1) 1, 4, 9, 16, 25, ...
(2) 2, 4, 6, 8, 10, 12, ...
(3) 1, -1, 1, -1, 1, -1, ...
(4) 1, -2, 4, -6, 10, -12, ...
(5) 1, 2, 1, 2, 1, 2, ...
(6) 1, 0, 3, 0, 5, 0, 7, ...
(7) ${\displaystyle {\frac {1}{2}},{\frac {3}{4}},{\frac {5}{6}},{\frac {7}{8}},...}$
(8) ${\displaystyle {\sqrt {2}},{\sqrt {5}},2{\sqrt {2}},{\sqrt {11}},...}$
(9) 1, 11, 111, 1111, 11111, ...
(10) 0.1, 0.11, 0.111, 0.1111, ...

### 遞推式的簡單尋找

A. ${\displaystyle a_{n}=\left\{{\begin{array}{l}1\quad (n=1)\\a_{n+1}+n-1\quad (n\geq 2,n\in \mathbb {N} )\end{array}}\right.}$
B. ${\displaystyle a_{n}=\left\{{\begin{array}{l}1\quad (n=1)\\a_{n-1}+n\quad (n\geq 2,n\in \mathbb {N} )\end{array}}\right.}$
C. ${\displaystyle a_{n}=\left\{{\begin{array}{l}1\quad (n=1)\\a_{n-1}+n-1\quad (n\geq 2,n\in \mathbb {N} )\end{array}}\right.}$
D. ${\displaystyle a_{n}=\left\{{\begin{array}{l}1\quad (n=1)\\a_{n+1}+n+1\quad (n\geq 2,n\in \mathbb {N} )\end{array}}\right.}$

(1) 1, 0.9, 0.81, 0.729, ...
(2) 1, 1, 2, 3, 5, 8, 13, ...
(3) ${\displaystyle 1,{\frac {1}{2}},{\frac {1}{3}},{\frac {1}{4}},{\frac {1}{5}},{\frac {1}{6}},...}$
(4) 1, 11, 111, 1111, 11111, ...
(5) 1, 12, 123, 1234, 12345, ...
(6) 1, 1.1, 1.11, 1.111, 1.1111, ...

①x坐標和y坐標都是不大於n的整數；②x坐標和y坐標至少有一個剛好等於n。

## 補充習題

• 求下列各數列的一個通項公式：
(1)${\displaystyle {\frac {1}{2}},{\frac {3}{4}},{\frac {7}{8}},{\frac {15}{16}},{\frac {31}{32}},...}$
(2)${\displaystyle -1,{\frac {8}{5}},-{\frac {15}{7}},{\frac {24}{9}},...}$