Arithmetic Progression: Formula for the n-th Term
1. Definition of an Arithmetic Progression (AP)
Definition: An arithmetic progression (AP) is a sequence of numbers in which each term, starting from the second, is obtained by adding the same constant number to the previous term. This constant is called the common difference and is denoted by d.
Formally, a sequence $a_1, a_2, a_3, \ldots$ is an arithmetic progression if and only if:
$$d = a_{n+1} - a_n \quad \text{for every } n \geq 1$$
In other words, the difference between any two consecutive terms is always the same number $d$.
✅ Example of an AP:
$3, 7, 11, 15, 19, \ldots$
Here $d = 7 - 3 = 4$, and every consecutive difference equals $4$.
❌ Not an AP:
$1, 4, 9, 16, 25, \ldots$
Differences: $3, 5, 7, 9, \ldots$ — they are not constant!
2. Types of Arithmetic Progressions
| Condition | Type | Example | Behavior |
|---|---|---|---|
| $d > 0$ | Increasing AP | $2, 5, 8, 11, 14, \ldots$ ($d = 3$) | Each term is larger than the previous one |
| $d < 0$ | Decreasing AP | $20, 17, 14, 11, 8, \ldots$ ($d = -3$) | Each term is smaller than the previous one |
| $d = 0$ | Constant AP | $5, 5, 5, 5, 5, \ldots$ ($d = 0$) | All terms are equal to $a_1$ |