A typical question in many textbooks, tests, exams, quizzes, and even IQ tests, is the following: Write down the next three terms in the following sequence: 2; 6; 12; ...

What number comes next?

Since there is a difference of 4 between the first two numbers and a difference of 6 between the next two numbers, it may seem 'reasonable' that one may '*guess*' that the next differences will be 8, 10 and 12, giving the next three numbers in the sequence as 20; 30; 42; ... However, if you thought the only
correct answer is 20; 30; 42; ... then you were WRONG as there are
INFINITELY many other possibilities!

For example, a cubic function (*p* -
20)*x*^{3}/6 + (21 - *p*)*x*^{2} + (11*p*
- 214)*x*/6 + 20 - *p* with *p* a variable, can be drawn
exactly through the first 3 points (indicated by points *A*, *B* and *C* in the graph above), but subsequent values of the
function can assume **any value** as *p* is changed. Drag point *P* to change the graph and its
function values.

(Note: The values of *p* in the slider above have, only for the sake of convenience, been restricted to integer values).

**Explore more**

1) Drag *p* until it equals 20 or 21. What do you notice?

2) What happens if *p* > 20? What happens if *p* becomes negative? Drag the point *P* far to the left until that happens.

3) Can you explain your observations above?

Generally, give a finite sequence of *n* numbers, infinitely
many polynomials of order *n* or higher can be drawn through the *n*
points. It is therefore complete mathematical NONSENSE in a test or
exam to ask what the next terms are - unless a SPECIFIC CONTEXT (e.g. a
figural pattern or some real world context) is also given that **uniquely
determines** the pattern.

For example, Duncan Samson in his paper "The Heuristic Beauty of
Figural Patterns" in *Learning and Teaching Mathematics*, no. 12,
2012, pp. 36-38, gives the following figural pattern for the
number pattern at the start that uniquely determines it as 2; 6; 12; 20; 30; 42; ...

Similarly, even though the original Rabbit problem formulated by Fibonacci is not a genuinely real world problem, it nonetheless uniquely determines the number pattern 1; 1; 2; 3; 5; ... Without the context, however, the first 5 Fibonacci numbers can be modelled exactly by infinitely many quintic or higher order polynomials; let alone infinitely many other types of functions.

Lastly, students need to be constantly reminded that an inductive pattern, unless one has a formal, structural proof for why it works, may break down as in the well-known example given below:

Copyright © 2019 KCP Technologies, a McGraw-Hill Education Company. All rights reserved.

Release: 2020Q3, semantic Version: 4.8.0, Build Number: 1070, Build Stamp: 8126303e6615/20200924134412

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Created by Michael de Villiers, April 2012; updated & converted to *WebSketchpad*, 22 Feb 2023.