An explicit formula is a mathematical expression that directly defines a term or value in a sequence or function based on its position or index. It provides a clear, concise way to calculate any term in the sequence without relying on previous terms.
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An explicit formula allows you to directly calculate any term in a sequence without needing to know the previous terms.
Explicit formulas are commonly used to represent arithmetic sequences, where each term is found by adding a constant difference to the previous term.
The general form of an explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
Explicit formulas are more efficient than recursive formulas when working with larger sequences, as they do not require calculating each term in order.
Sequence notation, such as $a_n$ or $a(n)$, is often used in conjunction with explicit formulas to represent the position of a term within the sequence.
Review Questions
Explain how an explicit formula differs from a recursive formula in the context of sequences.
An explicit formula directly defines a term in a sequence based on its position, whereas a recursive formula defines each term in a sequence based on the previous term(s). Explicit formulas are more efficient for calculating larger terms in a sequence, as they do not require computing each preceding term. In contrast, recursive formulas build up the sequence term-by-term, making them more suitable for smaller sequences or when the initial terms are known.
Describe the general form of an explicit formula for an arithmetic sequence and explain how it can be used to calculate any term in the sequence.
The general form of an explicit formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$, where $a_n$ represents the nth term, $a_1$ is the first term, and $d$ is the common difference between consecutive terms. This formula allows you to directly calculate the value of any term in the sequence based solely on its position $n$, without needing to know the previous terms. For example, if the first term is 3 and the common difference is 2, the 10th term can be calculated as $a_{10} = 3 + (10-1)2 = 21$.
Analyze how the use of sequence notation, such as $a_n$ or $a(n)$, contributes to the understanding and application of explicit formulas.
Sequence notation, such as $a_n$ or $a(n)$, is often used in conjunction with explicit formulas to represent the position of a term within the sequence. This notation clearly indicates that the formula is defining the $n$th term, allowing for a more concise and unambiguous representation. The use of subscripts or parentheses helps distinguish the variable $n$, which represents the term's position, from the actual value of the term itself. This notation reinforces the direct relationship between the term's position and its value, which is the essence of an explicit formula. By clearly identifying the term's position, the explicit formula can be applied to calculate any term in the sequence without the need for additional context or information.