Sequence (𝑢ₙ): An ordered list of numbers where each element is associated with an integer index (or rank) 𝑛. Formally, 𝑢ₙ is a function from natural numbers ℕ to real numbers ℝ, assigning each 𝑛 a real term.
Term of a Sequence: The value associated with a specific position 𝑛 in the sequence, denoted as 𝑢ₙ.
Index (𝑛): The position or rank of a term in the sequence, typically a natural number starting from 0 or 1.
Explicit Definition: A formula that directly computes the 𝑛-th term, e.g., 𝑢ₙ = 2𝑛 or 𝑣ₙ = 3𝑛² - 1.
Recurrence Relation: An equation expressing each term as a function of previous terms, e.g., 𝑢ₙ+1 = 3𝑢ₙ, where the next term depends on the current one.
A sequence is a list of numbers ordered by a specific rule, which can be explicit (direct formula) or recursive (relation involving previous terms).
The notation 𝑢ₙ often corresponds to a function 𝑢: ℕ → ℝ, with 𝑢(𝑛) = 𝑢ₙ.
Computing terms can be done directly using explicit formulas or iteratively via recurrence relations.
1. What is a sequence element in the context of sequences?
2. What is the primary role of an explicit formula in the context of sequences?
3. How do recurrence relations and explicit formulas differ in the process of generating sequence terms?
Explicit formula — role?
Directly computes the n-th term without previous terms.
Sequence — definition?
An ordered list of numbers indexed by natural numbers.
Recurrence relation — mechanism?
Defines each term based on previous terms.
Sequence — definition?
Ordered list of numbers with a rule.
Explicit formula — purpose?
Directly computes the $n$-th term.
Recurrence relation — role?
Defines a term based on previous ones.
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