geometric progression
Enable an agent to recognise a geometric progression, establish its parameters and conventions, assess its behaviour, and perform justified calculations or applications.
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 recognise a geometric progression, establish its parameters and conventions, assess its behaviour, and perform justified calculations or applications.
A geometric progression is a finite or infinite sequence whose successive terms satisfy a_{n+1} = r a_n for a fixed common ratio r, so that a_n = a_1 r^{n-1} for n ≥ 1, with the first term specified separately when r = 0.
It can be Test whether specified terms satisfy a fixed multiplicative recurrence and identify the transitions that fail.; Recover compatible initial terms and ratios while reporting ambiguity or inconsistency.; Generate terms, fill justified gaps, and solve threshold questions within a declared index range.; Classify magnitude, sign or phase behaviour, and sequence limits.; Calculate finite sums and justified infinite sums or truncation bounds.; Fit and assess a constant-factor approximation, then restrict extrapolation to its supported assumptions..
Distinguishing features
A single multiplier r must satisfy every defined adjacent transition; visually exponential growth alone is insufficient.
Constant consecutive quotients identify the multiplier only where the preceding term is nonzero; the recurrence can remain meaningful when quotient tests fail.
Under zero-based indexing, terms satisfy a_n = a_0 r^n for n ≥ 1, with a_0 specified separately to avoid ambiguity when r = 0.
A constant additive difference identifies an arithmetic progression; constant sequences can satisfy both definitions.
The progression is the sequence of terms, whereas its associated geometric series is a sum of those terms and has separate convergence conditions.
Scope
+ The recurrence a_(n+1) = r a_n and its relationship to a constant quotient of consecutive terms
+ Initial term, common ratio, index origin, numerical domain, and finite or infinite extent
+ Exact recognition, parameter recovery, and assessment of approximately geometric observations
+ Sign patterns, magnitude changes, term limits, and exceptional zero cases
+ Finite sums and conditions for convergence of the associated infinite series
+ Interpretation as a discrete constant-factor model with explicit assumptions
- Arithmetic progressions governed by a constant additive difference
- General recurrences with varying multipliers or additive forcing
- Continuous exponential functions except where sampled to produce a progression
- Geometric probability distributions and their statistical interpretation
- General series theory beyond sums associated with a geometric progression
- Domain-specific mechanisms of interest, population change, depreciation, or physical attenuation
Characteristics
- Initial term
- Real or complex value a_0; carries the units of the represented quantity when applicable Sets the scale and determines whether the sequence is identically zero.
- Common ratio
- Fixed real or complex multiplier r; dimensionless for terms representing the same quantity Determines multiplicative change, sign or phase behaviour, and convergence.
- Indexing and extent
- Index origin; consecutive index set; positive integer term count N or infinite extent Prevents off-by-one errors and distinguishes finite calculations from limit questions.
- Numerical and zero conventions
- Real or complex terms; recurrence or quotient definition; inclusion or exclusion of zero terms and zero ratio Makes recognition consistent in degenerate cases where common definitions can differ.
- Geometric fit status
- Exact, approximate under a specified tolerance, contradicted, or insufficient evidence Separates a proved sequence rule from a model inferred from limited observations.
- Parameter identifiability
- Unique, multiple compatible values, inconsistent, or undetermined Prevents an agent from reporting a unique ratio when zeros, missing terms, or sparse sampling leave ambiguity.
- Magnitude and sign or phase pattern
- Magnitude decreasing, constant, or increasing; fixed sign, alternating sign, complex phase evolution, or zero Supports correct behavioural descriptions without confusing magnitude growth with numerical monotonicity.
- Term and series convergence
- Recorded separately: convergent with limit or sum, divergent, or not applicable to finite extent A convergent sequence does not necessarily have a convergent associated series.
- Index-to-application mapping
- One index step corresponds to a declared period, distance increment, iteration, or other repeatable step Connects the multiplier to the actual spacing and conditions of an application.
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: 6 bundles · 11 layers · 16 findings · 31 questions.
Definition and sequence boundary Establishes exactly which sequence and definition the agent is using.
Zero cases, indexing, and confusion between a sequence and a sum can change whether an operation is valid.
Multiplicative membership
Tests membership through a fixed multiplier while exposing definition-dependent cases.
Fixed-ratio recurrence
The recurrence a_(n+1) = r a_n gives a direct membership test; quotient-based recognition additionally requires nonzero denominators.
- Does one fixed value of r satisfy every adjacent transition in the stated sequence? definition
- Does the adopted definition admit r = 0 and the all-zero sequence, and which reference establishes that convention? boundary
Index domain and extent
Records the first index, available transitions, and finite or infinite interpretation.
Declared sequence object
An ordered sequence with a declared index set must be distinguished from an unordered collection, a sampled subsequence, and an associated sum.
- What is the first index, how many terms are included, and are supplied terms consecutive? definition
- Is the object being assessed the progression itself, a finite sum, or an infinite series? boundary
Parameter recovery and evidence Determines which parameters the available terms actually support.
Sparse terms and zeros can leave several compatible ratios, while noisy measurements require a different standard from exact identities.
Exact parameter identification
Recovers parameters from indexed values without assuming uniqueness.
Ratio identifiability
A nonzero term followed by its successor fixes r; separated nonzero terms instead constrain a power of r and can leave root ambiguities.
- Which observed adjacent pair with a nonzero first term fixes the candidate ratio? measurement
- If only terms k steps apart are known, which solutions of r^k = a_(n+k)/a_n are permitted by the numerical domain? measurement
- Do all remaining terms support those candidates, including transitions into or out of zero? boundary
Observational support
Separates a stipulated or proved rule from a fitted pattern.
Exact versus approximate geometricity
A finite observed prefix can support a geometric fit but does not establish that unobserved terms follow the same rule.
- Is the fixed multiplier supplied by a definition, derived from a mechanism, or estimated from observations? provenance
- What measurement error, rounding rule, and residual tolerance justify calling the observations approximately geometric? measurement
- What evidence supports extending the fitted rule beyond the observed indices? action
Term behaviour and limits Characterises individual terms across the admissible range of ratios.
Positive growth, alternating signs, complex phase changes, and degenerate constants require different interpretations.
Magnitude, sign, and phase
Separates magnitude change from order and direction.
Ratio regimes
For nonzero terms, |r| governs magnitude change; negative real r alternates signs, while complex r also changes phase.
- Is |r| below, equal to, or above one, and is the initial term zero? measurement
- For real terms, how do the signs of a_0 and r affect sign alternation and monotonicity? measurement
- For complex terms, what phase increment replaces a real-valued increasing or decreasing description? boundary
Infinite term limits
Evaluates sequence convergence separately from summability.
Sequence limit classification
In real or complex arithmetic, a zero initial term gives the zero sequence; otherwise |r| < 1 gives limit zero, r = 1 gives a constant limit, and other ratios do not give a finite term limit.
- Which parameter case establishes a finite limit, unbounded magnitude, or bounded nonconvergence? measurement
- If terms tend to zero, has convergence of their sum been assessed independently? boundary
Sums and computation Supports aggregation and numerically reliable evaluation.
Finite sums, infinite sums, and floating-point calculations have distinct validity conditions.
Finite aggregation
Computes totals over a specified finite run of terms.
Finite sum cases
For N terms starting at a_0, S_N = a_0(1-r^N)/(1-r) when r ≠ 1, and S_N = N a_0 when r = 1.
- Which first term and term count define the requested sum, including both endpoint conventions? measurement
- Does r = 1 require the constant-sequence formula, or does r near one require a more stable numerical evaluation? action
Infinite sums and tail control
Checks convergence before assigning an infinite total or truncation guarantee.
Convergent sum and remainder
For a nonzero initial term, the infinite geometric series converges exactly when |r| < 1, with sum a_0/(1-r); after N terms its remainder is a_0 r^N/(1-r). The all-zero sequence sums to zero separately.
- Do the parameters justify an ordinary convergent infinite sum? boundary
- How many terms make the remainder magnitude no greater than the required error tolerance? action
- Could overflow, underflow, or cancellation materially affect the computed terms or total? measurement
Discrete modelling and permitted inference Connects geometric structure to applications and decisions.
A ratio only has an operational meaning relative to a step size and assumptions that sustain repeated multiplication.
Constant-factor interpretation
Records what one multiplication represents in the application.
Step and rate assumptions
For a positive real multiplier, r = 1 + g represents a fractional change g per declared step; unequal spacing or additive inputs require separate justification.
- What does one index increment represent, and are the relevant intervals or repeated operations comparable? definition
- What assumptions keep the multiplier fixed, and could additive inputs or changing conditions break the recurrence? boundary
Prediction and threshold actions
Restricts extrapolation and inverse calculations to supported parameter regimes.
Admissible term and index solutions
Predicted terms and threshold indices must respect integer indexing, sequence extent, parameter uncertainty, and the relevant sign or magnitude behaviour.
- Is the requested threshold about signed value or magnitude, and does the progression move toward it? action
- Which admissible integer index first satisfies the threshold, verified directly in the original recurrence or term formula? action
- How sensitive is the prediction to uncertainty in the initial term and common ratio over the proposed horizon? measurement
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.
- Definitions differ on whether zero initial terms and a zero common ratio are admitted; specify the convention before handling degenerate cases.
- For a nonzero initial term, an infinite geometric series converges exactly when |r| < 1; convergence of the progression itself is a separate question.
- This account uses recalled mathematical knowledge and does not represent source-verified research.
- Which of these check these first hold for the sense of geometric 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 geometric progression
- Infinite geometric progression
- Constant progression with common ratio 1
- Alternating-sign progression with negative real common ratio
- Geometrically decaying progression with 0 < |r| < 1
- Progression with complex terms and common ratio
- Which of these kinds and varieties hold for the sense of geometric 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.
- Modelling balances under a constant compound interest rate without deposits or withdrawals
- Modelling quantities that grow or decay by a fixed factor over equal time steps
- Constructing logarithmically spaced frequency bands and numerical grids
- Calculating repeated percentage reductions
- Analysing algorithms whose work or problem size changes by a fixed multiplicative factor
- Which of these real-world use hold for the sense of geometric 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 ratio r - No universal typical range; commonly real, and possibly complex - Dimensionless
- Initial term a_1 - Unrestricted by the recurrence; some definitions exclude zero - Same as the modelled quantity, or dimensionless
- Number of terms in a finite progression - Positive integer - Count
- Which of these typical measurements hold for the sense of geometric 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 geometric progression, which is a sequence, with a geometric series, which sums its terms.
- Using the infinite-sum formula a_1/(1-r) for a nonzero initial term when |r| ≥ 1.
- Applying the finite-sum formula a_1(1-r^N)/(1-r) at r = 1; the sum is then N a_1.
- Computing a common ratio by division when a preceding term is zero; the recurrence and quotient definitions need explicit conventions in this case.
- Assuming a constant growth factor remains valid indefinitely in empirical systems with changing rates or resource constraints.
- Which of these failure modes and hazards hold for the sense of geometric 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 progression - Successive terms have a constant additive difference rather than a constant multiplicative factor; constant sequences can satisfy both definitions.
- Geometric series - A geometric series is the sum of terms of a geometric progression, rather than the sequence itself.
- Exponential function - A geometric progression has a discrete index; an exponential function can have a continuous argument and yields a geometric progression when sampled at equal intervals.
- Harmonic progression - The reciprocals of its nonzero terms form an arithmetic progression, rather than its terms necessarily sharing a constant ratio.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of geometric progression this model covers, and on what evidence? provenance
What the second pass must settle
- Which reference convention should the registry adopt for zero ratios, all-zero sequences, and single-term progressions?
- Should the registered concept include complex-valued progressions directly, or use a real-valued default with an explicit extension?
- Does an existing Vercy world model already own this concept or the sequence-versus-series boundary?
- Which application-specific error models and tolerances should govern claims that measured data are approximately geometric?
- Which reference sources should establish the model's definitions, exceptional cases, and numerical evaluation guidance?