Basic mathematical questions connected with each task or problem are the questions whether or not a solution exists and, should one exist, whether this solution is unique. Of course, these questions of existence and uniqueness of solutions may not be ignored when tasks are utilized in psychological contexts. In this paper the problems of solvability and (non-)uniqueness are analyzed for a certain type of number sequence task that is widely used in psychometrics as well as in other fields of psychological research and application.
The solvability and uniqueness problems of number sequence tasks cannot be discussed comprehensively as long as the domain of permissible rules for the sequence is not definitely made explicit. Following this guideline, in this article the class of linear-recursive number sequences is defined (Section 2) and analyzed. This class of number sequences turns out to be quite comprehensive; it contains, for example, all arithmetic and geometric sequences and, on the whole, a large proportion of number sequences ocurring in common psychometric tests on intelligence and ability. On the other hand, linear-recursive sequences represent only a tiny segment of a universe of sequences that might be regarded as ''regular'' as far as anyone can judge.
In Section 3 the problems of solvability and uniqueness are explicated for a general type of task concerning linear-recursive number sequences. Section 4 adapts the theory of linear equation systems in order to derive several criteria and an obvious decision-making procedure for the problems of existence and uniqueness of recursion formulae for number sequences. Sections 5 - 7 deal with the analysis of the uniqueness problem in connection with a specific type of task based on linear-recursive number sequences with strong constraints for the domain of permissible recursion formulae.
The solution-analysis of tasks including linear-recursive number sequences reveals, essentially, two levels of non-uniqueness:
Most striking in our analysis is the discovery that the worst cases of non-unique solvability can occur even in strongly restricted types of number sequence tasks. Despite strong restrictions for permitted recursion formulae, we have, for instance, observed types of number sequences where permissible recursion formulae of various degrees always exist, and where this non-uniqueness of solution does not depend on the length of the given sequence and, thus, cannot be prevented by prolonging the given sequence. Furthermore, we have detected example sequences where no solution of the existing ones can be considered as the ''simplest'' one in a psychological sense, whereas each solution formula produces a different extension of the given sequence.
Undoubtedly, certain types of number sequences provoke a worst case scenario for number sequence tasks in each psychological context. The fact that those ugly number sequence types exist even under strong restrictions for the permitted sequence rules is highly disturbing. This offers an insight that is hardly in accordance with Jensen's standpoint cited in Section 1, and should prevent the careless use of sequence tasks in psychological application.
In psychometric tests, task instructions for number sequence tasks often provide some example rules accounting for the presented number sequence, but, in general, they do not explicitly limit the domain of permissible rules. In contrast to this common application of number sequence tasks, the results of our analysis confirm the requirement that number sequence tasks should be presented only accompanied by a definite instruction about the domain of permissible sequence rules; otherwise, the solvability conditions of number sequence tasks cannot be rationally analyzed and controlled, and consequently the use of number sequence tasks as reliable and valid instruments of psychological measuring or knowledge modeling must be questioned.
Fortunately, within the class of linear-recursive number sequences the problems of solvability and non-uniqueness of solution can be brought under control by several means as, in particular, selection of appropriate restrictions for the permissible recursion formulae, careful construction of each applied number sequence task, and rational task analysis involving the provided mathematical instruments. Taking into account the basic problems demonstrated in this paper and utilizing the proposed analytic instruments, specific types of linear-recursive number sequences could turn out to be appropriate prototypes for the frequently utilized sequence tasks.