- Write the terms of a sequence defined by an explicit formula and a recursive formula
- Use factorial notation
Sequences Defined by an Explicit Formula
The Main Idea
- Definition of a Sequence:
- A function whose domain is a subset of positive integers
- Each number in the sequence is called a term
- Types of Sequences:
- Finite: Has a specific number of terms
- Infinite: Continues indefinitely (uses ellipsis …)
- Explicit Formula:
- Defines the nth term using its position in the sequence
- Typically written as [latex]a_n = [\text{ expression involving }n][/latex]
- Representation Methods:
- List of terms: [latex]{a_1, a_2, a_3,...,a_n,...}[/latex]
- Table of values
- Graph (discrete points)
- Piecewise Sequences:
- Different formulas for different ranges of [latex]n[/latex]
[latex]{a_{n}}=\begin{cases}2n^{3} & \text{if }n\text{ is odd} \\[1mm] \dfrac{5n}{2} & \text{if }n\text{ is even}\end{cases}[/latex]
Finding an Explicit Formula
The Main Idea
- Pattern Recognition:
- Look for relationships between terms and their positions
- Analyze numerators and denominators separately for fraction sequences
- Identify patterns in signs, exponents, or bases
- Alternating Sequences:
- Use [latex](-1)^n[/latex] or [latex](-1)^{n-1}[/latex] to represent alternating signs
- Don’t rearrange terms in numerical order
- General Structure:
- [latex]a_n = [\text{ expression involving }n][/latex]
- May involve algebraic, exponential, or trigonometric functions
- Verification:
- Always test your formula for the first few terms of the sequence
[latex]\{9;−81,729;−6,561;59,049\}[/latex]
[latex]\left\{-\dfrac{3}{4},-\dfrac{9}{8},-\dfrac{27}{12},-\dfrac{81}{16},-\dfrac{243}{20},\dots\right\}[/latex]
[latex]\left\{\dfrac{1}{{e}^{2}}, \dfrac{1}{e}, 1, e, {e}^{2},...\right\}[/latex]
[latex]{a}_{n}=\dfrac{4n}{{\left(-2\right)}^{n}}[/latex]
You can view the transcript for “Finding the formula of alternating signs of a sequence” here (opens in new window).
Sequences Defined by a Recursive Formula
The Main Idea
- Definition of Recursive Formula:
- Defines each term using preceding term(s)
- Must state initial term(s)
- Structure:
- Initial condition(s): [latex]a_1 = [\text{value}][/latex], ([latex]a_2 = [\text{value}][/latex], if needed)
- Recursive rule: [latex]a_n = [\text{expression involving }a_{n-1}, a_{n-2},\text{ etc.}][/latex]
- Applications:
- Useful for sequences defined by step-by-step processes
- Effective for modeling natural growth and decay phenomena
- Comparison with Explicit Formulas:
- Recursive: Defines terms relative to previous terms
- Explicit: Defines terms directly based on their position
- Limitations:
- May be computationally intensive for large [latex]n[/latex]
- Not always easy to find a specific term without calculating all preceding terms
[latex]\begin{align}{a}_{1}&=2\\ {a}_{n}&=2{a}_{n - 1}+1\text{, for }n\ge 2\end{align}[/latex]
You can view the transcript for “Ex: Finding Terms in a Sequence Given a Recursive Formula” here (opens in new window).
Factorial Notation
The Main Idea
- Definition of Factorial:
- [latex]n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1[/latex], for [latex]n \geq 1[/latex]
- [latex]0![/latex] is defined as [latex]1[/latex]
- Properties:
- Factorials grow very quickly
- Often used in combinatorics and probability
- Application in Sequences:
- Can appear in both explicit and recursive formulas
- Often leads to rapidly converging or diverging sequences