Difference Method

*Quadratic sequences* of numbers are characterized by the fact that the ** difference between terms always changes by the same amount**.

Consequently, the

Here's what we mean, consider the

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

If a *sequence* is *quadratic* then its *formula* can be written:
\[u_n = an^2+bn+c\]
For example, the *sequence*, we saw above: \(6,11,18,27,38,51 \dots \) has formula:
\[u_n = n^2 + 2n + 3 \]
Indeed, if we replace \(n\) by (for example) \(1\) and \(2\) we'll find the first and second terms of the *sequence*, that's:
\[\begin{aligned} u_1 & = 1^2 + 2\times 1 + 3 \\
&= 1 + 2 +3 \\
u_1 & = 6 \end{aligned} \]

Given the *first few terms of a quadratic sequence*, we find its *formula*
\[u_n = an^2 + bn +c\]
using the following *three equations*:
\[\begin{cases}
2a = \text{second difference}\\
3a + b = \text{difference between the first two terms, \(u_2 - u_1\)}\\
a+b+c = \text{the first term, \(u_1\)}
\end{cases}\]
The following illustration shows all of the *differences* we're referring to in these equations, for the *quadratic sequence*: \(6,11,18,27,38,51 \dots \)

Looking at this, and the *formula* we saw above, each of the *equations* is:
\[\begin{cases}
2a = 2\\
3a + b = 5\\
a+b+c = 6
\end{cases}\]
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 stated here, we can find each of the *three coefficients* \(a\), \(b\) and \(c\).

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

In the following tutorial we review *how to find the formula, for the \(n^{\text{th}}\) term of a quadratic sequence* .

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

- The sequence whose first few terms are: \[-3,0,5,12,21,32, \dots \]
- The sequence whose first few terms are: \[-4,2,12,26,44,66,92, \dots \]
- The sequence whose first few terms are: \[2,6,12,20,30,42,56, \dots \]
- The sequence whose first few terms are: \[1,0,-3,-8,-15,-24,-35, \dots \]
- The sequence whose first few terms are: \[3,7,13,21,31,43,57, \dots \]

- \(u_n = n^2 - 4\)
- \(u_n = 2n^2 - 6\)
- \(u_n = n^2 + n\)
- \(u_n = -n^2+2n\)
- \(u_n = n^2 + n + 1\)