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Cubic Sequences
Difference Method

Cubic sequences of numbers are characterized by the fact that the third difference between its terms is constant.

For example, consider the sequence: \[4,14,40,88,164, \dots \] looking at the first, second and third difference of this sequence would look like:

Looking at this we can see that the third difference is constant, and not equal to zero, this means it is a cubic sequence.

Formula for the \(n^{\text{th}}\) term

If a sequence is cubic then its formula can be written: \[u_n = an^3+bn^2+cn + d\] For example, the sequence, we saw above: \(4,14,40,88,164, \dots \) has formula: \[u_n = n^3 + 2n^2 - 3n + 4 \] Indeed, if we replace \(n\) by (for example) \(1\) and \(2\) we'll find the first and second terms of the sequence, that's: \[\begin{aligned} u_1 & = 1^3 + 2\times 1^2 - 3 \times 1 + 4 \\ &= 1 + 2 - 3 + 4 \\ u_1 & = 4 \end{aligned} \]

and
\[\begin{aligned} u_2 &= 2^3 + 2\times 2^2 - 3 \times 2 + 4 \\ & = 8 + 2\times 4 - 6 + 4 \\ & = 8+8-6+4 \\ u_2 & = 14 \end{aligned} \] We learn how to find the formula for the \(n^{\text{th}}\) term below.

Method - Finding the formula for the \(n^{\text{th}}\) term

Given the first few terms of a cubic sequence, we find its formula \[u_n = an^3+bn^2+cn +d\] using the following four equations: \[\begin{cases} 6a = \text{third difference} \\ 12a + 2b = \text{1st second difference} \\ 7a + 3b + c = \text{difference between the first two terms} \\ a + b + c + d = \text{first term} \end{cases}\] This can be a little confusing at first so we illustrate it here with the cubic sequence: \[4,14,40,88,164, \dots \] The following illustration shows all of the differences we're referring to:


Looking at this, and our the formula we saw above, each of the four equations would be: \[\begin{cases} 6a = 6 \\ 12a + 2b = 16 \\ 7a + 3b + c = 10 \\ a + b + c + d = 4 \end{cases}\] We can see that the values that we use, the ones we've boxed in the illustration, are always the first on each row.

Using each of these equations, in the order they're given here, we can find each of the four coefficients \(a\), \(b\), \(c\) and \(d\).


This is best explained in the following tutorial, watch it now.

Tutorial

In the following tutorial we learn how to use the four equations to find the formula for the \(n^{\text{th}}\) term of a cubic sequence.

Exercise

Find the formula for the \(n^{\text{th}}\) term of each of the following sequences:

  1. The sequence whose first few terms are: \[4,14,40,88,164, \dots \]

  2. The sequence whose first few terms are: \[4,23,66,145,272, \dots \]

  3. The sequence whose first few terms are: \[-1,1,-5,-25,-65, \dots \]

  4. The sequence whose first few terms are: \[1,14,65,178,377, \dots \]

  5. The sequence whose first few terms are: \[11,6,-25,-100,-237, \dots \]

Answers Without Working

  1. \(u_n = n^3+2n^2-3n+4\)

  2. \(u_n = 2n^3+5n-3\)

  3. \(u_n = -n^3+2n^2+3n-5\)

  4. \(u_n = 4n^3-5n^2+2\)

  5. \(u_n = -3n^3+5n^2+n+8\)