# How do you find the pattern for a list?

Feb 13, 2018

A few thoughts...

#### Explanation:

I think you are referring to some kind of sequence, but it is not clear what sort of problems you want to address.

Let's look at a few sequence types and their associated rules.

Arithmetic sequence

An arithmetic sequence has a common difference between terms. So to get from one term to the next you need to add the common difference.

We can write a recursive rule:

${a}_{n + 1} = {a}_{n} + d$

where $d$ is the common difference.

We also need to specify the starting point, the first term $a$:

${a}_{1} = a$

We can write the explicit rule for a general term as:

${a}_{n} = a + d \left(n - 1\right)$

Geometric sequence

A geometric sequence has a common ratio between terms. So to get from one term to the next you need to multiply by the common ratio.

We can write a recursive rule:

${a}_{n + 1} = {a}_{n} \cdot r$

where $r$ is the common ratio.

We also need to specify the starting point:

${a}_{1} = a$

We can write the explicit rule for a general term as:

${a}_{n} = a {r}^{n - 1}$

Linear recursion

A sequence may also be defined using a linear recursive rule, where each successive term is based on the two (or more) previous terms.

The classic example of such a sequence is the Fibonacci sequence, definable recursively by:

${F}_{0} = 0$

${F}_{1} = 1$

${F}_{n + 2} = {F}_{n + 1} + {F}_{n}$

It starts:

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , \ldots$

How do you get an explicit rule for a linear recursion like this?

Consider a geometric sequence:

$1 , x , {x}^{2} , {x}^{3} , \ldots$

If it satisfies the linear recursive rule for $1 , x , {x}^{2}$, then it will continue to satisfy it.

So given a rule:

${a}_{n + 2} = p {a}_{n + 1} + q {a}_{n}$

we can associate the quadratic equation:

${x}^{2} = p x + q$

Calling the two roots of this equation $\alpha$ and $\beta$ note that any sequence given by:

${a}_{n} = A {\alpha}^{n} + B {\beta}^{n}$

will satisfy the recursive rule.

We then just have to choose $A$ and $B$ so that the first two terms match the initial two terms of the sequence.

In the case of the Fibonacci sequence, we find:

${F}_{n} = \frac{1}{\sqrt{5}} \left({\varphi}^{n} - {\varphi}^{- n}\right)$

where $\varphi = \frac{1}{2} + \frac{1}{2} \sqrt{5}$