## What number comes next?

Write down the next three terms in the following
sequence: 2; 6; 12; ...

####

What number comes next?

If you thought the only
correct answer is 20; 30; 42; ... then you were WRONG as there are
INFINITELY many possibilities!

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

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.

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Modified by Michael de Villiers, 20 April 2012.