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Cubic sequences

Cubic sequences

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A cubic sequence is a sequence with only a constant 3rd difference. A 1st difference is the difference between one term of the sequence and the next, a 2nd difference is the difference between 2 adjacent 1st differences, and a 3rd difference is the difference btween 2 adjacent 2nd differences (see the diagram of an example cubic sequence).

Basics of cubic sequences

To find the next terms of a given cubic sequence, one needs to figure out the constant 3rd difference, and then use that difference to find the next 2nd difference (by adding it to the previous 2nd difference), which can be used to find the next 1st difference. Once the next first 1st difference is known, one can add this to the last known term to find the next term in the sequence, as shown in the example.

Terms in a cubic sequence

A cubic sequence has the general form of anΒ³ + bnΒ² + cn + d, where a is non zero because a cubic sequence has to have a nΒ³ present. Consider a table of the 1st to 3rd differences of this general cubic sequence. This gives us a set of equations for a, b, c and d (shown in red). So to find the nth term of a cubic sequence one just needs to find the 1st, 2nd and 3rd differences, and use the equations to find what the coefficients a, b, c and d of the general form would be.

nth term of cubic sequences

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