等差數列 Arithmetic Sequence (AS)

公式 Formula

(i) 通項 General Term
　\( T_n = a + (n - 1)d \)
‧ \( T_n \) = 第n項 \( n^{\text{th}} \) term
‧ \( a \) = 首項 First Term
‧ \( d \) = 公差 Common Difference

(ii) \( T_n - T_{n-1} = d \)

(iii) $T_n = \frac{1}{2} \left( T_{n-1} - T_{n+1} \right)$

(iv) 首 \( n \) 項之和 Sum of the first \( n \) terms
　　= $\frac{n}{2}(T_1 + T_n)$

　　或 or

　　= $\frac{n}{2}[2a + (n-1)d]$

例子 Example

考慮等差數列 Consider the arithmetic sequence 1, 4, 7, 10, ...

首項 First Term \( (a) \) = 1

公差 Common Difference \( (d) \) = 4 - 1 = 3

通項 General Term = $T_n = 1 + (n-1)\cdot 3 = 1 + 3n - 3 = 3n - 2$

第20項 The 20th term $T_{20} = 1 + (20-1)3 = 58$

首 20項之和 Sum of the first 10 terms
=$\frac{20}{2}\cdot(1+58)$

或or

$\frac{20}{2}\big[2\cdot 1 + (20-1)\cdot 3\big]$

= 590