Definition and General Form of an AP
An Arithmetic Progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number $d$ (common difference) to the preceding term, except the first term $a$.
$$\text{General Form: } a, \, a+d, \, a+2d, \, a+3d, \, \dots, \, a+(n-1)d$$
- First term $= a$
- Common difference $= d = a_{k+1} - a_k$ (can be positive, zero, or negative)
Formulas Bank
1. $n$-th Term of an AP ($a_n$):
$$a_n = a + (n - 1)d$$
$n$-th term from the end:
$$a_n' = l - (n - 1)d \quad (l = \text{last term})$$
2. Sum of First $n$ Terms ($S_n$):
$$S_n = \frac{n}{2} [2a + (n - 1)d] = \frac{n}{2} (a + l)$$
3. Key Property:
$$a_n = S_n - S_{n-1}$$
