Quadratic and cubic patterns

Compare a sequence with familiar powers of its position. Constant first differences suggest a linear rule, constant second differences a quadratic rule, and constant nonzero third differences a cubic rule. Differences guide the search; check the resulting rule against all supplied terms.

Example

The terms \(3,10,29,66,127\) are two more than \(1,8,27,64,125\). Thus the stated cube pattern has \(T_n=n^3+2\), giving \(T_6=218\).

A visual pattern may justify the rule more directly: an n by n by n stack has \(n^3\) cubes, so adding two extra cubes gives \(n^3+2\). A finite list without an intended pattern does not determine a unique infinite sequence.