# How are patterns used to create algebraic expressions?

Jun 20, 2015

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 ${n}^{t h}$ 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...

$1 , 4 , 10 , 20 , 35 , 56$
$\to 3 , 6 , 10 , 15 , 21$
$\to 3 , 4 , 5 , 6$
$\to 1 , 1 , 1$

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...

$0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34$
$\to 1 , 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13$
$\to - 1 , 1 , 0 , 1 , 1 , 2 , 3 , 5$
$\to 2 , - 1 , 1 , 0 , 1 , 1 , 2$
$\to - 3 , 2 , - 1 , 1 , 0 , 1$
$\to 5 , - 3 , 2 , - 1 , 1$
...

But it is possible to find a formula for ${F}_{n}$, viz

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

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

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