**Klaus Korossy**

**Psychologisches Institut der Universität Heidelberg**

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**Abstract**

This article deals with the problems ofexistence and uniquenessof solution in connection withnumber sequence tasks(often called ''number-series completion tasks'') that are widely used in various psychological contexts. It is argued that the problems of existence and uniqueness of solution can be rationally analyzed and controlled only when each type of sequence task applied is explicitly referred to some domain of permissible rules accounting for the included sequence type. Following this guideline, the class oflinear-recursive number sequence tasksis introduced where the sequence members are related by linear recursive equations. The investigation of this type of sequence task can substantially profit from the theory oflinear equation systems. The analysis carried out uncovers several types of non-uniqueness of solution. Most striking is the fact that worst cases of non-uniqueness may occur even in strongly restricted subtypes of linear-recursive number sequences as is shown for a specific type of task. The results of the analyses suggest that (number) sequence tasks should not be applied in psychological contexts if not accompanied by an instruction that refers explicitily to some domain of permitted sequence rules.

Key words:number series -- number-series completion tasks -- linear-recursive number sequences -- existence and uniqueness of solution

- Number sequence tasks in psychometrics and cognitive psychology
- Linear-recursive number sequences: definitions and examples
- General type of task, solvability and uniqueness of solution
- Existence and uniqueness of solution formulae
- A special type of linear-recursive number sequence tasks
- Analysis of the uniqueness problem for the special type of number sequence task
- Decidability of the uniqueness problem for the special type of number sequence task
- Summary and general discussion
- References
- Author's Address

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