# Patterns and Expressions

## Key Questions

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.

• That would vary on what is meant on "sum", "difference" and "product". Other than that exception, sum, difference, product, and quotient are just fancy words for adding, subtracting, multiplying, and dividing respectively.
There are the simple symbols: $a + b , a - b , a \times b , a \div b$ (or $\frac{a}{b}$).
There is a special symbol for difference used in some math and science equations: $\Delta x$
This means there is a final value and an initial $x$ value. You would simply subtract the final and the initial to get the change or difference.

This is used in the equation to find the slope of a line:
$\frac{\Delta y}{\Delta x}$

Is the same as

$\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
This means you subtract y-coordinate points and x-coordinate points on a line to find the slope.

There is also a special symbol for summing and products, and it can get a little confusing:

${\sum}_{n = 0}^{10} n$

This is the symbol for summing a function of $n$ denoted as a capital sigma
The bottom number denoted as $n$ is the starting number.
The top number is the ending number.
You then plug in $n$ for each number up to 10 and add them up.
The answer to the summing operation above is 55.

${\prod}_{n = 1}^{10} n$

This is the symbol for product denoted as a capital pi (this is NOT $3.14159265 \ldots$ pi, that's lowercase) . The same rules for summing apply to products, but you multiply instead of add. The answer to the above product is 3,628,800.

That is also the answer to 10! Note that $n$ starts at 1 and not 0 in the product.

As for a special quotient symbol, I'm not 100% sure if such a thing exists.

Algebraic expressions are formed of integer constants and variables. They follow algebraic operations such as addition, subtraction, division, and multiplication.

#### Explanation:

$2 x \left(3 - x\right)$ is an algebraic expression in factorised form. Another example is $\left(x + 3\right) \left(x + 10\right)$.

Algebraic expressions can also have powers (indices): $\left({x}^{2} + 3\right) {x}^{3}$

The expressions have multiple variables as well: $x y \left(2 - x\right)$

Etc.