How do you find the pattern for a list?

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(n-1)#

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 * 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 = ar^(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,...#

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

Consider a geometric sequence:

#1, x, x^2, x^3,...#

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 = px+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 = 1/sqrt(5)(varphi^n - varphi^(-n))#

where #varphi = 1/2+1/2sqrt(5)#