63.4 Arithmetic Recursive Rule
The Arithmetic Recursive Rule defines a sequence where each term is generated by adding a constant difference to the previous term.
Arithmetic Recursive Rule is a two-part description of an arithmetic sequence that states the value of the first term directly and then defines every later term in relation to the term immediately before it, mirroring the step-by-step way the sequence is actually generated.
Arithmetic Initial-Term Statement
Stating the Starting Value
The recursive rule begins with a direct statement of the first term's value, since a recursive definition requires a known starting point before it can generate anything further.
Why an Initial Statement Is Required
Without this starting value stated explicitly, the recursive part of the rule would have nothing to build from, since every later term is defined only in terms of the one before it, tracing back eventually to this first term.
Previous-Term Reference
Referring to the Immediately Preceding Term
The recursive rule refers to the term immediately before the one being defined, using a position number one less than the term currently being described.
Why Only the Immediately Preceding Term Is Used
An arithmetic sequence's defining property concerns only the relationship between consecutive terms, so the recursive rule needs no reference to any term further back than the one directly preceding it.
Recursive Difference Addition
Adding the Common Difference
The recursive rule generates the next term by adding the common difference to the value of the previous term.
Why This Addition Step Is the Core of the Rule
This single addition step is the direct algebraic expression of the sequence's defining property, that each term differs from the one before it by exactly the same constant amount.
Arithmetic Recursive Rule Construction
Assembling the Two-Part Definition
The complete recursive rule combines the initial-term statement with the previous-term addition step, together forming a full definition of the sequence.
Why Both Parts Are Necessary Together
Neither part alone is sufficient: the initial term without the addition rule provides no way to generate further terms, and the addition rule without the initial term has no starting value to build upon.
Recursive Term Generation
Building the Sequence Step by Step
Using the completed recursive rule, terms are generated one at a time, each calculated from the value of the term found immediately before it.
Why This Process Requires Sequential Calculation
Because each term's calculation depends directly on knowing the value of the term before it, this generation process must proceed sequentially from the first term onward, unlike the explicit rule, which can reach any term directly.
Explicit-Recursive Agreement
Confirming Both Rules Describe the Same Sequence
The terms produced by applying the recursive rule step by step are compared against the terms produced by the explicit rule at the same positions, confirming that both rules describe the identical sequence.
Why This Agreement Should Always Hold
Because both rules are built from the same first term and the same common difference, they represent two different ways of expressing the same underlying pattern, and any disagreement between them indicates an error in how one of the two rules was constructed.
Recursive Starting Condition Check
Confirming the Initial Term Matches the Given Sequence
This check confirms that the stated initial term in the recursive rule exactly matches the first term of the sequence as it was originally given, before the recursive rule is used to generate any further terms.
Why This Check Comes First
Because every term generated by the recursive rule ultimately depends on this single starting value, confirming its correctness before generating any further terms prevents an error at the very foundation of the rule from propagating through the entire generated sequence.