Cubic Sequences (Leaving Cert Mathematics): Revision Notes
Cubic Sequences
Introduction
A cubic sequence is a sequence of numbers where the third difference of every consecutive term is constant.
Consider a sequence of the first 7 cubed numbers :
The first difference of the sequence forms a quadratic sequence, the second difference forms a linear (arithmetic) sequence and the third difference is constant.
The general term of a cubic sequence is given by :
A useful property of cubic sequence is that the third difference is given by :
Example
Find the general term, of the following cubic sequence -1,13,51,125,247,...
First, identify the third common difference. Start by taking the first difference.
Then the second difference :
Finally, the third common difference :
Apply the property of the coefficient, :
The general term looks like this at the moment :
Substitute some of the terms that known to form three equations in terms of .
With three equations established, solve simultaneously. Refer to the simultaneous equations chapter.