Many number sequences have the characteristic property that subsequent members are related to the preceding members by linear equations.
2.1 Examples.
Number sequences like those shown in the examples above, which are widely used in psychometric tests and psychological experimentation, will be of central interest in this article. In this section we first introduce some basic concepts concerning the special type of number sequences shown in the examples. The following abbreviations will be used:
the set of natural numbers (without 0)
the set of rational numbers
the set of n-tupel of elements of set A
2.2 Definition.
Let be a (infinite) non-constant sequence
of rational numbers. If for a given
there exists an (r+1)-tupel
with such that the recursive equations
are satisfied, then the sequence is called
linear-recursive of degree r with the recursion
coefficients and the constant c,
in brief -recursive (or -recursive).
The (r+1)-tupel is called a recursion formula
(of degree r) for , occasionally a solution
to (1); (which may be a proper subset of ) is called the
domain for the recursion formulae of degree r.
Remarks.
(i) Within the calculus of differences, (1) describes an inhomogeneous linear difference equation with constant coefficients and constant inhomogenity (see e.g. Markuschewitsch, 1955; Rommelfanger, 1986, pp. 36-37), where, however, it is permitted that the recursion equation (1) may be satisfied beginning only with the members of a higher index than r+1. Because our consideration differs from the typical way of looking at the problems in the calculus of difference, we will not pursue that approach here.
(ii) With respect to the types of number sequences applied in psychometrics or experimental psychology we presuppose that the members of the considered sequences, the recursion coefficients, and the constant c are elements of the field IQ of rational numbers which is closed under the elementary arithmetic operations. Special attention will later be given to the domain for the recursion formula.
(iii) In Definition 2.2 the condition is to prevent that for instance the sequence , with for , may be classified as -recursive. According to 2.2 this sequence is not -recursive, since the condition is not satisfied. However, this sequence is also not -recursive of degree 1, because the recursion equation does not hold for (see Remark (i)). Instead the sequence is, for instance, -recursive of degree 4, because holds for .
(iv) It might seem reasonable to include constant number sequences in the concept of linear-recursive number sequences. We will refrain from doing so in this paper because the rather trivial and uninteresting case of a constant number sequence would demand, mainly because of notational reasons, special consideration in some proofs.
Examples. In the examples of 2.1, the sequence (a) is -recursive as well as -recursive; the sequence (b) is -recursive; the Fibonacci sequence (c) is -recursive; the sequence (d) is -recursive.
As is seen in the examples above certain sequences can be represented by several different recursion formulae. On the other hand of course, each recursion formula describes infinitely many sequences depending on the given initial members.
Perhaps the concept of a linear-recursive number sequence seems to be relatively specific. However, a large proportion of number sequences used in psychometric tests can be conceived as linear-recursive sequences although they may not appear to be so at first glance. The following examples may serve to verify this.
2.3 Examples.
Up until now we have been dealing with infinite sequences. However, common number sequence tasks applied in psychological testing and research present only a finite sequence of some numbers and demand regular continuation of this sequence. We denote such a finite n-term sequence by and extend the definition of infinite linear-recursive sequences in 2.2 to n-term sequences.
2.4 Definition. Let be a finite, non-constant n-term sequence of rational numbers. Suppose that for some the equations of form (1) for hold for some . Then also is said to be -recursive (resp. -recursive) (of length n). The other notions of 2.2 are used accordingly.
The usual number sequence tasks demand that on the basis of a recognized regularity the number sequence is to be ''continued'' or ''completed'' by determining one or more subsequent members of the sequence. To make this demand more precise we introduce the notion of an ''extension'' of a given finite (linear-recursive) sequence.
2.5 Definition. Let the n-term sequence for some be -recursive with some . If for the p-term sequence is also -recursive and if the initial n members coincide with , then is called the k-fold -extension of .
2.6 Examples.
Note at this point a serious problem uncovered by 2.6 (d): The ninth member of the sequence, 14, suggested by the onefold -extension is not found among the multiple choice answers provided in the test (KFT, Form B, Item 32). Actually, the sequence could also be continued by 12, which is suggested by the sequence rule apparently intended by the test constructors. In this paper the problems of non-unique solutions to number sequence tasks are extensively discussed within the class of linear-recursive number sequences.