Essential Concepts
Sequences and Their Notations
Sequence Basics
- A sequence is a function whose domain is the set of positive integers (1, 2, 3…)
- Each number in the sequence is called a term
- Terms are denoted as [latex]a_1, a_2, a_3, \ldots, a_n, \ldots[/latex] where [latex]a_1[/latex] is the first term, [latex]a_2[/latex] is the second term, and [latex]a_n[/latex] is the [latex]n[/latex]th term
- A finite sequence has a limited number of terms; an infinite sequence continues indefinitely
- The ellipsis ([latex]\ldots[/latex]) indicates the sequence continues
Explicit Formulas
- An explicit formula defines the [latex]n[/latex]th term of a sequence using the position [latex]n[/latex]
- Allows you to find any term directly without finding previous terms
- Written in the form [latex]a_n = f(n)[/latex]
- To find terms, substitute [latex]n = 1, 2, 3, \ldots[/latex] into the formula
Alternating Sequences
- Sequences where terms alternate in sign (positive, negative, positive, negative, etc.)
- Use [latex](-1)^n[/latex] if the first term is negative
- Use [latex](-1)^{n-1}[/latex] if the first term is positive
Recursive Formulas
- A recursive formula defines each term using one or more preceding terms
- Must always state the initial term(s)
Arithmetic Sequences
- An arithmetic sequence has a constant difference between consecutive terms
- The common difference [latex]d[/latex] is found by subtracting any term from the next term: [latex]d = a_n - a_{n-1}[/latex]
- Explicit Form: The [latex]n[/latex]th term is given by [latex]a_n = a_1 + (n-1)d[/latex]
- Recursive Form: Written as [latex]a_n = a_{n-1} + d[/latex] for [latex]n \geq 2[/latex]. Requires stating the initial term [latex]a_1[/latex]
Geometric Sequences
- A geometric sequence has a constant ratio between consecutive terms
- The common ratio [latex]r[/latex] is found by dividing any term by the previous term: [latex]r = \frac{a_n}{a_{n-1}}[/latex]
- Explicit Form: The [latex]n[/latex]th term is given by [latex]a_n = a_1 r^{n-1}[/latex]
- Recursive Form: Written as [latex]a_n = r \cdot a_{n-1}[/latex] for [latex]n \geq 2[/latex]
Series and Summation Notation
- A series is the sum of the terms in a sequence
- Summation notation uses the Greek letter sigma ([latex]\Sigma[/latex]) to represent sums
- Format: [latex]\sum_{k=1}^{n} a_k[/latex] means sum [latex]a_k[/latex] from [latex]k=1[/latex] to [latex]k=n[/latex]
- [latex]k[/latex] is the index of summation
- The lower limit is the starting value; the upper limit is the ending value
Arithmetic Series
- The sum of the terms of an arithmetic sequence
- Formula for the sum of the first [latex]n[/latex] terms: [latex]S_n = \frac{n}{2}(a_1 + a_n)[/latex]
Geometric Series
- The sum of the terms of a geometric sequence
- Formula for the sum of the first [latex]n[/latex] terms: [latex]S_n = \frac{a_1(1-r^n)}{1-r}[/latex] where [latex]r \neq 1[/latex]
Infinite Geometric Series
- The sum exists only when [latex]-1 < r < 1[/latex]
- Formula for the sum: [latex]S = \frac{a_1}{1-r}[/latex] when [latex]-1 < r < 1[/latex]
- If [latex]|r| \geq 1[/latex], the series diverges (sum does not exist)
Key Equations
| [latex]n[/latex]th term of an arithmetic sequence | [latex]a_n = a_1 + (n-1)d[/latex] |
| Recursive formula for arithmetic sequence | [latex]a_n = a_{n-1} + d, \, n \geq 2[/latex] |
| [latex]n[/latex]th term of a geometric sequence | [latex]a_n = a_1 r^{n-1}[/latex] |
| Recursive formula for geometric sequence | [latex]a_n = r \cdot a_{n-1}, \, n \geq 2[/latex] |
| Sum of first [latex]n[/latex] terms of arithmetic series | [latex]S_n = \frac{n}{2}(a_1 + a_n)[/latex] |
| Sum of first [latex]n[/latex] terms of geometric series | [latex]S_n = \frac{a_1(1-r^n)}{1-r}, \, r \neq 1[/latex] |
| Sum of infinite geometric series | [latex]S = \frac{a_1}{1-r}, \, -1 < r < 1[/latex] |
Glossary
arithmetic sequence
A sequence where the difference between consecutive terms is always the same (constant difference [latex]d[/latex]).
common difference
The constant value [latex]d[/latex] added to each term to get the next term in an arithmetic sequence, calculated as [latex]d = a_n - a_{n-1}[/latex].
common ratio
The constant value [latex]r[/latex] by which each term is multiplied to get the next term in a geometric sequence, calculated as [latex]r = \frac{a_n}{a_{n-1}}[/latex].
diverges
A series diverges when its sum is not a real number or does not approach a finite value.
ellipsis
The symbol ([latex]\ldots[/latex]) used to indicate that a sequence or pattern continues indefinitely.
explicit formula
A formula that defines the [latex]n[/latex]th term of a sequence using the position [latex]n[/latex] in the sequence.
finite sequence
A sequence with a limited number of terms.
geometric sequence
A sequence where the ratio between consecutive terms is always the same (constant ratio [latex]r[/latex]).
index of summation
The variable (often [latex]k[/latex], [latex]i[/latex], or [latex]n[/latex]) used in summation notation to represent the position of terms being summed.
infinite sequence
A sequence that continues indefinitely without ending.
infinite series
The sum of all terms in an infinite sequence.
lower limit of summation
The starting value of the index in summation notation.
[latex]n[/latex]th term of the sequence
The general term of a sequence, denoted [latex]a_n[/latex], representing any term based on its position [latex]n[/latex].
partial sum
The sum of the first [latex]n[/latex] terms of a series, denoted [latex]S_n[/latex].
recursive formula
A formula that defines each term of a sequence using one or more preceding terms. Must include initial term(s).
sequence
A function whose domain is the set of positive integers; an ordered list of numbers following a specific pattern.
series
The sum of the terms in a sequence.
sigma
The Greek letter [latex]\Sigma[/latex] used in summation notation to indicate a sum.
summation notation
A notation using the sigma symbol ([latex]\Sigma[/latex]) to represent the sum of terms in a series.
term
An individual number in a sequence.
upper limit of summation
The ending value of the index in summation notation.