# How are patterns used to create algebraic expressions?

##### 1 Answer

#### Answer:

This question is rather general, so I will only address a small fragment of it...

#### Explanation:

How do you identify patterns and express them as algebraic expressions?

For example, given the sequence

What is the pattern? How do you find the next number in the sequence? What is the formula for the

With these sort of problems, it is often helpful to construct a sequence of differences between successive terms, repeating this process to see if you end up with a constant sequence...

Since it has taken

We can directly construct the formula for

I rather like this discrete variant on Taylor's theorem.

This approach will not work well with the Fibonacci sequence, since it is essentially exponential, not polynomial...

...

But it is possible to find a formula for

where

Even the area of numeric sequences is too big to give a full answer.