1. Finite and Infinite Sequences
Sequences can be grouped by the number of terms they contain. Some sequences stop after a certain point, while others continue without end. Understanding this distinction helps identify the nature and behavior of the pattern.
1.1. Finite Sequences
A finite sequence contains a limited number of terms. It has a definite beginning and a definite ending. For example:
\( 4, 8, 12, 16, 20 \)
This sequence stops at the fifth term, so it is finite.
1.2. Infinite Sequences
An infinite sequence continues endlessly. It does not have a final term. For example:
\( 1, 3, 5, 7, 9, \ldots \)
The dots indicate that the sequence goes on forever.
2. Monotonic Sequences
Monotonic sequences move consistently in one direction. They may always increase or always decrease. This steady behavior makes them easier to analyze.
2.1. Increasing Sequences
An increasing sequence is one where each term is greater than or equal to the term before it. Example:
\( 2, 5, 5, 7, 10, \ldots \)
Here, the numbers never go down.
2.2. Decreasing Sequences
A decreasing sequence is one where each term is less than or equal to the previous term. Example:
\( 20, 15, 10, 10, 5, \ldots \)
Each term stays the same or moves downward.
2.3. Strict vs Non-Strict Monotonicity
Strictly monotonic sequences change at every step:
- Strictly increasing: \( 1, 2, 3, 4, \ldots \)
- Strictly decreasing: \( 10, 9, 8, 7, \ldots \)
Non-strictly monotonic sequences allow repeated terms:
- Non-strictly increasing: \( 2, 4, 4, 6, 6, \ldots \)
- Non-strictly decreasing: \( 5, 5, 3, 3, 1, \ldots \)
3. Bounded Sequences
A sequence is bounded if its terms stay within a certain range. In other words, the terms never grow beyond a limit or fall below a limit.
3.1. Bounded Above
A sequence is bounded above if there is a number that every term is less than or equal to. This number is called an upper bound.
Example:
\( 1, 3, 4, 5, 5, 4, 3 \)
This sequence is bounded above by 5.
3.2. Bounded Below
A sequence is bounded below if all its terms are greater than or equal to some fixed number, called a lower bound.
Example:
\( 7, 6, 6, 8, 9 \)
This sequence is bounded below by 6.
3.3. Bounded Both Sides
If a sequence is bounded above and bounded below at the same time, it is said to be bounded on both sides.
Example:
\( 3, 5, 4, 6, 5, 4 \)
This sequence lies between 3 and 6, so it is bounded on both sides.
4. Periodic Sequences
A periodic sequence repeats its terms in a fixed cycle. After a certain number of steps, the sequence returns to the same pattern and continues repeating.
4.1. Definition of Periodicity
A sequence is periodic if there exists a positive integer \(k\) such that:
\( a_{n+k} = a_n \) for every \( n \)
The number \(k\) is called the period of the sequence.
4.2. Examples of Periodic Sequences
Example 1:
\( 1, 2, 3, 1, 2, 3, 1, 2, 3, \ldots \)
The pattern repeats every 3 terms, so the period is 3.
Example 2 (trigonometric):
\( \sin(n) \) produces a repeating sequence when \(n\) is measured in certain steps.