Question

How are patterns used to create algebraic expressions?

Answer 1

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 `1, 4, 10, 20, 35, 56`

What is the pattern? How do you find the next number in the sequence? What is the formula for the `n^(th)` term in the sequence?

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...

`1, 4, 10, 20, 35, 56`
`-> 3, 6, 10, 15, 21`
`-> 3, 4, 5, 6`
`-> 1, 1, 1`

Since it has taken `3` steps to get to a constant sequence, the original sequence is expressible as a cubic expression.

We can directly construct the formula for `a_n` from the first term of each of these sequences as:

`a_n = color(red)(1) + color(red)(3) * n/(1!) + color(red)(3) * (n(n-1))/(2!) + color(red)(1) * (n(n-1)(n-2))/(3!)`

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...

`0, 1, 1, 2, 3, 5, 8, 13, 21, 34`
`-> 1, 0, 1, 1, 2, 3, 5, 8, 13`
`-> -1, 1, 0, 1, 1, 2, 3, 5`
`-> 2, -1, 1, 0, 1, 1, 2`
`-> -3, 2, -1, 1, 0, 1`
`-> 5, -3, 2, -1, 1`
...

But it is possible to find a formula for `F_n`, viz

`F_n = (phi^n - (-phi)^-n)/sqrt(5)`

where `phi = (1+sqrt(5))/2`

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