arithmetic progression
Enable an agent to recognize arithmetic progressions, establish their parameters and validity, and determine which computations and transformations are justified.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to recognize arithmetic progressions, establish their parameters and validity, and determine which computations and transformations are justified.
An arithmetic progression is a finite or infinite sequence of numbers in which the difference between each term and its predecessor is constant, so that a_n = a_1 + (n − 1)d for a fixed common difference d.
It can be Verify constant differences over a complete finite sequence or prove them from a defining rule.; Recover an anchor and common difference from sufficient indexed information, reporting ambiguity or inconsistency.; Generate terms, interpolate missing indexed terms, and extrapolate when the defining rule is established or explicitly assumed.; Determine whether a candidate value occurs and identify its admissible indices.; Compute finite block sums and solve endpoint or term-count constraints.; Apply affine value transformations, contiguous restrictions, or regularly spaced index sampling and determine the resulting progression..
Distinguishing features
For every adjacent pair in the stated index range, a_(n+1) - a_n equals the same value d; a constant ratio alone does not establish this condition.
The explicit term rule is a_n = a_k + (n-k)d for integer indices in the progression's domain, distinguishing the sequence from a continuous line.
Where triples exist, a_(n+2) - 2a_(n+1) + a_n = 0; nonlinear polynomial sequences generally fail this test.
Ordering and index gaps matter: unequal differences between observations may still fit a progression when the observations are not consecutive.
A constant sequence qualifies with d = 0 and can also qualify as geometric, so these classes are not disjoint.
Scope
+ Constant-difference identity, including increasing, decreasing, and constant progressions
+ First term, common difference, index origin, and finite or infinite extent
+ Explicit, recursive, and partially observed representations
+ Recovery and consistency checking of parameters from indexed terms or constraints
+ Term membership, finite sums, and justified transformations
- General sequences whose consecutive differences are not constant
- Geometric progressions defined by a constant ratio
- Arithmetic series as independently studied summation objects, especially convergence theory
- Arithmetic functions and number-theoretic properties not implied by constant spacing
- Empirical trend models and statistical estimation beyond assessing compatibility with an arithmetic progression
- Continuous affine functions except as representations restricted to integer indices
Characteristics
- Value domain
- Integers, rationals, or reals in the default model; other domains require explicit conventions Determines which terms, differences, divisions, and comparisons are admissible.
- Anchor term
- Value a_k at a recorded integer index k; dimensionless or in the terms' stated unit Together with the common difference and index domain, fixes the progression.
- Common difference
- Exact value d in term units per index increment Controls spacing, direction, term generation, and arithmetic identity.
- Index domain
- Finite consecutive integer interval, one-sided infinite interval, or explicitly supported two-sided interval Determines which terms exist and whether an endpoint or finite total is available.
- Term count
- Nonnegative integer N for a finite representation, or infinite; short-sequence conventions must be stated Determines how strongly observations constrain the difference and which finite-sum operations apply.
- Verification state
- Proved on the declared domain, compatible with observed terms, contradicted, or underdetermined Prevents a finite pattern from being treated as proof of an unrestricted continuation.
- Representation
- Explicit rule, recurrence with initial condition, complete finite list, or partial indexed observations Determines how the progression can be evaluated and what evidence supports its identity.
- Observation indexing
- Mapping from each observed value to its sequence index, including known or unknown gaps Supports correct parameter recovery when observed terms are not consecutive.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 5 bundles · 9 layers · 14 findings · 22 questions.
Arithmetic identity Establishes what constant difference means for the declared values and index domain.
An agent must distinguish a valid progression from a suggestive list and handle degenerate cases explicitly.
Constant-difference condition
Connects the adjacent-difference invariant with explicit and recursive definitions.
Equivalent term rules
On a consecutive integer index domain, a_(n+1) = a_n + d and a_n = a_k + (n-k)d describe the same progression when an anchor is fixed.
- What anchor, common difference, and index domain define this progression? definition
- Does the supplied rule establish the same adjacent difference at every admissible step? measurement
Domain and degenerate cases
Records the numerical setting and cases in which familiar tests become insufficient.
Constant and short sequences
Zero difference gives a constant progression. Two real-valued terms determine a difference, while a singleton does not determine one; acceptance of empty or singleton progressions requires a stated convention.
- Which convention governs empty and singleton sequences, and is the common difference supplied or recoverable? boundary
- Is d zero, and which operations therefore need a separate constant-sequence case? action
Representation and evidence Records how terms are specified and the strength of the evidence for arithmetic structure.
Correct calculations depend on index interpretation, while justified continuation depends on evidence beyond a short prefix.
Index-aware representation
Makes index origin, extent, and observation gaps explicit.
Indexed observations
Values must be attached to their actual indices; a gap of h indices implies a value difference of hd, not necessarily d.
- Are the listed terms consecutive, regularly sampled, or attached to irregularly spaced known indices? definition
- What source establishes the index origin, omitted positions, and final index if one exists? provenance
Proof and observation
Separates exact arithmetic identity from compatibility with limited or rounded data.
Verification strength
A complete finite list can be checked exhaustively, but a finite prefix alone does not prove an infinite continuation. Rounded observations require an explicit error model before arithmetic compatibility can be assessed.
- Is arithmetic structure established by a definition or proof, by a complete finite check, or only suggested by observations? provenance
- Are values exact, and if they are rounded or measured, what uncertainty bounds govern the compatibility test? measurement
Parameter recovery and membership Determines which progression fits the available constraints and which values belong to it.
Agents need to resolve missing terms and candidate membership without silently assuming uniqueness or valid indices.
Constraint identifiability
Uses indexed values and additional conditions to recover or reject parameters.
Recovering the difference
For distinct known indices i and j over the reals, d = (a_j-a_i)/(j-i). Further observations must satisfy that same rule, and integer-valued progressions impose additional integrality constraints.
- Do the available indexed terms determine a unique d, or do unknown index gaps leave multiple possibilities? measurement
- Do all remaining observations and value-domain restrictions agree with the recovered anchor and difference? boundary
Candidate-term membership
Tests values against both the term equation and the allowed index range.
Admissible index test
For d ≠ 0, a candidate x belongs exactly when n = k + (x-a_k)/d is an integer in the index domain. For d = 0, every existing term equals the anchor.
- Does the candidate produce an integer index inside the declared finite or infinite index domain? measurement
- If d is zero, does the candidate equal the anchor, and which existing indices should be returned? action
Derived operations and preservation Governs finite aggregation and transformations that retain constant spacing.
Recognizing a progression should enable reliable operations while exposing their preconditions and scope.
Finite block aggregation
Computes sums using the correct endpoints and number of terms.
Finite sum and endpoints
A nonempty block of N consecutive terms has sum N(first + last)/2, with last = first + (N-1)d. The formula describes a finite block and does not supply a total for an infinite progression.
- What are the inclusive starting and ending indices, and does N equal their difference plus one? measurement
- When solving from a sum or endpoint, do the resulting term counts and indices satisfy the required integer and range constraints? action
Arithmetic-preserving transformations
Tracks how affine value changes and regular index selection alter the parameters.
Transformed spacing
An affine value map b_n = αa_n + β gives difference αd. Sampling a_(r+hm) gives difference hd wherever the sampled indices exist; reversing a finite progression changes the difference to -d.
- What anchor, difference, and valid index domain result from the proposed transformation? action
- Is the selected index spacing constant, or does preservation require a special case such as d = 0? boundary
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- This describes the standard numerical sense; the core definition is broadly settled.
- Some number-theoretic contexts require a nonzero or positive common difference to exclude trivial progressions.
- The definition permits real or complex terms; increasing and decreasing classifications apply to real-valued progressions.
- Which of these check these first hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Finite arithmetic progression
- Infinite arithmetic progression
- Increasing real arithmetic progression (d > 0)
- Decreasing real arithmetic progression (d < 0)
- Constant arithmetic progression (d = 0)
- Which of these kinds and varieties hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Representing equally spaced sampling times or measurement positions.
- Calculating totals when successive payments or deposits increase by a fixed amount.
- Describing regularly spaced seats, objects or construction elements.
- Studying patterns among integers and primes in number theory.
- Generating evenly spaced numerical grids.
- Which of these real-world use hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Common difference d - Any real number for a real-valued progression; no universal typical range - Same unit as the terms, or dimensionless
- Number of terms N - Any positive integer for a nonempty finite progression; infinite progressions have no finite term count - Dimensionless count
- Which of these typical measurements hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Confusing a constant additive difference with the constant multiplicative ratio of a geometric progression.
- Introducing an indexing error by mixing a_0-based and a_1-based formulas.
- Confusing the sequence with its arithmetic series, which is a sum of terms.
- Assuming that a few equally spaced observations establish an arithmetic pattern for all later terms.
- Using exact equality of floating-point differences to test approximately measured or computed data.
- Which of these failure modes and hazards hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Arithmetic sequence - Usually a synonym for arithmetic progression, with no mathematical distinction.
- Arithmetic series - A series sums the terms of an arithmetic progression; the progression is the sequence itself.
- Geometric progression - Successive terms are related by a fixed multiplicative factor rather than a fixed additive difference.
- Affine function - An arithmetic progression is an affine function of an integer index; an affine function can have a broader domain.
- Polynomial sequence - An arithmetic progression has constant first differences; sequences generated by higher-degree polynomials generally require higher-order differences to become constant.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of arithmetic progression this model covers, and on what evidence? provenance
What the second pass must settle
- Which registry or world-model entry already owns the broader sequence concept, and does an existing publication already cover arithmetic progression?
- Which reference convention should govern empty and singleton progressions and the distinction between a finite arithmetic sequence and a finite segment of an infinite progression?
- Should complex-valued, modular, or abstract additive-group progressions be supported here, and which domain-specific tests would replace real-valued division and order?
- Should two-sided infinite progressions be first-class instances or explicitly declared extensions of the default indexing convention?
- What evidence and uncertainty conventions should govern recognition from rounded observations without treating approximate agreement as exact arithmetic identity?