Difference Method

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

For example, consider the

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

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:
\[\begin{aligned} u_1 & = 1^3 + 2\times 1^2 - 3 \times 1 + 4 \\
&= 1 + 2 - 3 + 4 \\
u_1 & = 4 \end{aligned} \]

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.

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*.

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

- The sequence whose first few terms are: \[4,14,40,88,164, \dots \]
- The sequence whose first few terms are: \[4,23,66,145,272, \dots \]
- The sequence whose first few terms are: \[-1,1,-5,-25,-65, \dots \]
- The sequence whose first few terms are: \[1,14,65,178,377, \dots \]
- The sequence whose first few terms are: \[11,6,-25,-100,-237, \dots \]

- \(u_n = n^3+2n^2-3n+4\)
- \(u_n = 2n^3+5n-3\)
- \(u_n = -n^3+2n^2+3n-5\)
- \(u_n = 4n^3-5n^2+2\)
- \(u_n = -3n^3+5n^2+n+8\)