Increasing and Decreasing Sequences
What is a Sequence?
A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. We denote the terms as $a_1, a_2, a_3, \ldots, a_n$, where $a_n$ represents the $n$-th term.
📈 Increasing Sequences
Definition: A sequence $\{a_n\}$ is increasing if each term is greater than the previous term:
$$a_{n+1} > a_n \quad \text{for all } n \geq 1$$
Equivalently, the difference between consecutive terms is always positive:
$$a_{n+1} - a_n > 0$$
Examples:
- $2, 5, 8, 11, 14, \ldots$ (each term increases by 3)
- $1, 2, 4, 8, 16, \ldots$ (each term doubles)
- $1, 4, 9, 16, 25, \ldots$ (perfect squares: $n^2$)
Strictly vs. Non-Strictly Increasing:
If $a_{n+1} \geq a_n$ (allowing equal terms), we call it non-decreasing or weakly increasing. When $a_{n+1} > a_n$ strictly, it's strictly increasing.
If $a_{n+1} \geq a_n$ (allowing equal terms), we call it non-decreasing or weakly increasing. When $a_{n+1} > a_n$ strictly, it's strictly increasing.
📉 Decreasing Sequences
Definition: A sequence $\{a_n\}$ is decreasing if each term is less than the previous term:
$$a_{n+1} < a_n \quad \text{for all } n \geq 1$$
Equivalently, the difference between consecutive terms is always negative:
$$a_{n+1} - a_n < 0$$
Examples:
- $20, 17, 14, 11, 8, \ldots$ (each term decreases by 3)
- $100, 50, 25, 12.5, \ldots$ (each term halves)
- $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots$ (the sequence $\frac{1}{n}$)
Strictly vs. Non-Strictly Decreasing:
If $a_{n+1} \leq a_n$ (allowing equal terms), we call it non-increasing or weakly decreasing. When $a_{n+1} < a_n$ strictly, it's strictly decreasing.
If $a_{n+1} \leq a_n$ (allowing equal terms), we call it non-increasing or weakly decreasing. When $a_{n+1} < a_n$ strictly, it's strictly decreasing.