Step 2: Number Sequences, Term Indexing & Common Difference
The mathematical definition of a sequence provides the language of discrete mathematics and algorithmics.
1. The Anatomy of a Sequence
- Sequence: An ordered collection of numbers $a_1, a_2, a_3, \dots, a_n$ mapped from positive integers $\mathbb{N} \to \mathbb{R}$.
- Terms: The individual elements $a_k$, where $k$ is the term index or rank.
- Finite vs Infinite Sequences:
- Finite: Has a terminating last term $a_n$ (e.g. Day temperatures of November: 30 terms).
- Infinite: Continues indefinitely (e.g. Set of all primes: $2, 3, 5, 7, 11, \dots$).
2. Definition of an Arithmetic Progression (AP)
A sequence $a_1, a_2, a_3, \dots$ is an Arithmetic Progression if and only if the difference between any consecutive pair of terms is constant:
$$a_{k+1} - a_k = d \quad (\forall k \ge 1)$$
where $d$ is the Common Difference.
3. Classification by Common Difference ($d$)
- Increasing AP ($d > 0$):
- Example: $4, 9, 14, 19, 24, \dots$ ($a = 4, d = +5$).
- Decreasing AP ($d < 0$):
- Example: $50, 43, 36, 29, \dots$ ($a = 50, d = -7$).
- Constant AP ($d = 0$):
- Example: $\pi, \pi, \pi, \pi, \dots$ ($a = \pi, d = 0$).
4. Verifying if a Sequence is an AP
To test if sequence $x_n$ is an AP, compute $x_{n+1} - x_n$. If the result is independent of $n$ (a constant number), it is an AP; if it depends on $n$, it is NOT an AP!
- Example 1: $a_n = 3n + 7 \implies a_{n+1} - a_n = [3(n+1) + 7] - [3n + 7] = 3$ (Constant $\implies$ AP with $d = 3$).
- Example 2: $b_n = n^2 + 1 \implies b_{n+1} - b_n = [(n+1)^2 + 1] - [n^2 + 1] = 2n + 1$ (Depends on $n \implies$ NOT an AP).