Difference Method

*Linear sequences* of numbers are characterized by the fact that to get from one term to the next we *always add the same amount*.

The amount we add is known as the ** difference**, frequently called the

For example, the sequences: \[3,7,11,15,19,23, \dots \]

For the first, we always add \(3\) and for the second we always add \(-2\).

Here we illustrate this further for the first

We can see quite clearly that we're *always adding the same amount to get from one term to the next*, this means it is a ** linear sequence**.

If a *sequence* is *linear* then its *formula* can be written:
\[u_n = an+b\]
For example, the *sequence* whose *first few terms* are:
\[3,7,11,15,19,23, \dots \]
has formula:
\[u_n = 4n - 1\]
This lets us calculate any *term of the sequence* directly.

For instance to calculate the \(4^{\text{th}}\) term we would replace \(n\) by \(4\) in our formula:
\[\begin{aligned} u_4 &= 4\times 4 - 1 \\
& = 16 - 1 \\
u_4 &= 15
\end{aligned}\]
We learn how to find the *formula for the \(n^{\text{th}}\) term below*.

Given the *first few terms* of a *linear sequence*, we find its *formula*
\[u_n = an+b\]
using the following *two equations*:
\[\begin{cases}
a = \text{difference between first two terms} \\
a+b = \text{first term of the sequence}
\end{cases}\]
To illustrate this further, we look at our *sequence* \(3,7,11,15,19,23, \dots\).

For this *sequence* these equations tell us:

Looking at this, and the *two equations* we saw above, each of the equations is:
\[\begin{cases}
a = 4 \\
a+b = 3
\end{cases}\]
Using these *two equations*, we can find each of the *coefficients* \(a\) and \(b\).

The method for using this *formula* is illustrated in the *tutorial* below.

In the following *tutorial* we review the method for *findng the formula for the \(n^{\text{th}}\) term of a linear sequence*. Watch it now.

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

- The sequence whose first few terms are: \[2,6,10,14,18, \dots \]
- The sequence whose first few terms are: \[13,10,7,4,1, \dots \]
- The sequence whose first few terms are: \[-3,4,11,18,25, \dots \]
- The sequence whose first few terms are: \[3,3.5,4,4.5,5, \dots \]
- The sequence whose first few terms are: \[12, 5, -2, -9, -16, \dots \]

- \(u_n = 4n-2\)
- \(u_n = -3n + 16\)
- \(u_n = 7n - 10\)
- \(u_n = 0.5n +2.5\)
- \(u_n = -7n + 19\)