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 advantage of using an explicit formula to define a sequence?
3. What is the primary role of an explicit formula in the context of sequences?
Explicit formula — role?
Directly computes the n-th term without previous terms.
Sequence — definition?
Ordered list of numbers with a rule.
Sequence — definition?
An ordered list of numbers indexed by natural numbers.
Explicit formula — purpose?
Directly computes the $n$-th term.
Recurrence relation — mechanism?
Defines each term based on previous terms.
Recurrence relation — role?
Defines a term based on previous ones.
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