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 x3*(p - 20)/6 + x2*(21 - p) + x*(11p - 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.