Arithmetic Progressions (Arithmetic Sequences)
A rigorous yet accessible treatment — from definitions to applications
1. Definition and Notation
What is an Arithmetic Progression?
A sequence of real numbers $a_1, a_2, a_3, \ldots$ is called an arithmetic progression (AP) if there exists a fixed real number $d$ — called the common difference — such that every consecutive pair satisfies
$$ a_{n+1} - a_n = d \qquad \text{for every } n \ge 1. $$Equivalently, the general form of an AP is
$$ a,\; a+d,\; a+2d,\; a+3d,\; \ldots $$Recursive Definition
Specify the first term and a rule that generates each subsequent term from the previous one:
$$ a_1 = a, \qquad a_{n+1} = a_n + d. $$This tells you how the sequence grows but requires you to compute every preceding term to reach $a_n$.
Explicit (Closed-Form) Definition
A direct formula that gives any term by its index:
$$ a_n = a_1 + (n-1)d. $$This lets you jump straight to the 1000th term without computing the first 999.
Key Points about $d$
- $d > 0$: the sequence is increasing (e.g., $3, 7, 11, 15, \ldots$).
- $d < 0$: the sequence is decreasing (e.g., $20, 17, 14, 11, \ldots$).
- $d = 0$: every term is the same — a constant sequence (e.g., $5, 5, 5, 5, \ldots$).
Examples and Non-Examples
| Sequence | AP? | Reason |
|---|---|---|
| $2, 5, 8, 11, 14, \ldots$ | Yes | $d = 3$ (constant difference) |
| $10, 6, 2, -2, -6, \ldots$ | Yes | $d = -4$ |
| $7, 7, 7, 7, \ldots$ | Yes | $d = 0$ |
| $1, 2, 4, 8, 16, \ldots$ | No | Differences are $1, 2, 4, 8, \ldots$ — not constant (geometric progression) |
| $1, 1, 2, 3, 5, 8, \ldots$ | No | Fibonacci — differences are $0, 1, 1, 2, 3, \ldots$ |
2. General Term Formula — Derivation
The Formula
$$ \boxed{a_n = a_1 + (n-1)d} $$Derivation by Iteration (Telescoping)
Write out the recurrence step by step:
$$ a_2 = a_1 + d $$
$$ a_3 = a_2 + d = a_1 + 2d $$
$$ a_4 = a_3 + d = a_1 + 3d $$
$$ \vdots $$
$$ a_n = a_1 + (n-1)d $$