Sequences are *ordered lists* of numbers that are **always progressing toward** a certain goal. These numbers can be negative or positive, so long as they continue to increase or increase and then decrease, you have a sequence.

Another common sequence is Roman numerals. These start with I and go all the way up to V then ascend again with VII, VIII, IX… All the way up to XIX then XX then LX then LXX then LXXX then LCXC then CC.

These are both very simple sequences and easy to recognize but there are many more complicated ones.

## These are some common sequences

Sequences are a pattern of numbers that follow a specific order. These numbers can grow in value, decrease in value, or stay the same as the next number is added.

Many sequences can be found in everyday life. The seven-*day week sequence continues* with the next day of the week after you work or attend school. The ten-**digit phone number sequence continues** with the addition of a country code and personal phone number.

The sixteen-**digit social security number sequence continues** with the addition of your birth date. The twenty-eight-**day calendar sequence continues** with the addition of a new day.

There are so many sequences in life that it is impossible not to find one.

## The golden ratio

In mathematics, the golden ratio is a **number often described** as the ratio of 1 to 1.6. This is the same ratio that you can find in nature, in plants and seashells, for example.

Many artists use this ratio when creating their artwork. They will often use it for the proportion of different parts of a painting or sculpture. For example, the width of the canvas may be *one golden ratio wider* than the length of the painting or sculpture.

In art, this ratio can create a sense of harmony and balance within the piece. It can also make your **eye travel back** and forth within the artwork, making it more engaging.

## Pascal’s triangle

A special triangle called Pascal’s triangle is related to the binomial theorem. Pascal’s triangle contains numbers that are arranged in rows and columns, with each number in the column being the sum of the two numbers above it in the row.

The first row and column only *contain one number*, which is 1. Every other cell contains a sum of two numbers, so every other cell in the triangle contains a new number.

By looking at any given cell in the triangle, you can determine what numbers are above it and what their sum is. For example, looking at the very first cell on the **top left corner**, you can see that there are two 1s above it and their sum is also 1.

Looking at the next cell to its right, there are two 2s above it and their sum is also 2. Looking at any *given cell reveals* this information about cells above it.

## Fibonacci sequence

Another sequence that can be used to answer what’s next questions is the Fibonacci sequence. Named after mathematician Leonardo Fibonacci, this sequence starts with zero and one, and then adds the *previous two numbers* to get the next number in the series.

All numbers in the sequence are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… so how do you figure out the next number? You can’t just add the *last two numbers —* you have to think about what comes before that number.

For example: What’s the next number in the sequence? 28. How do you get there? You have to think about what comes before 28 — that would be 27. So the next number in the sequence is 29. What is 16 again? Well 15 comes before it so you have to go back one more step in order to get back to 16.

## Zeno’s dichotomy

A *famous paradox named* for the Greek philosopher Zeno claims that motion is impossible. Why? Because, he claims, to reach any destination you have to pass through the middle, but to pass through the middle you have to pass through the middle, and so on!

Zeno’s ideas about motion were very influential in ancient Greece. But **later philosophers showed** that his ideas were flawed, and most agree that he didn’t really understand the concept of motion.

But what if we replaced “motion” with a different word? What if we said that “the next number in the sequence is 7”? Would it be impossible for there to be a next number in the sequence? No! We can prove it!

Well, sort of.

## Binomial distribution function

The *binomial distribution function* is used to determine the probability of getting a certain number of successes in a given number of trials.

This function is used in cryptography to determine the probability of a correct guess on a password attempt, **password guessing software uses** this function. It is also used in betting and gambling to determine the expected value of money one will win based on the bets they make.

Cryptography uses a simple version of this function that considers only getting zero or one successes in a given trial as a success. This makes it more suited for *password guessing programs* as it will not **consider repeated attempts** as successes.

Whether you are betting on horse races or gambling at casinos, this function determines the expected value of money one will win based on the bets they make. It takes into consideration the likelihood of winning based on the type of bet made.

## Gaussian distribution function

Another way of understanding the *normal distribution function* is by thinking about average values. The Gaussian distribution function describes what the *average value would* be for any given measurement, or how many things of a given size there are in a collection.

For example, think about how many people in the world have brown hair, and consider that to be one measurement. Then, think about the average length of people’s hair, and consider that to be another measurement.

The number of people with brown hair who have short hair is the same as the number of people with *long blonde hair*. On average, there is no difference in length between these hairs – they are both average hairs.

Similarly, on average, there are an equal number of people with brown hair who have medium-length hair. These all fall within the normal distribution.

## Normal distribution function

Something important to understand about Z-scores is that they represent a relative position in a distribution, not an absolute value.

A person who is **1 standard deviation** above the *average income level* in the U.S. would be very wealthy, but on the other hand, that person would be quite average in the world population.

So, what does it mean if someone has a Z-score of 2.5? It means that person’s test score is 2.*5 standard deviations* above the average score. It does not mean that person is exceptionally intelligent or talented; it just means that person scored higher than most people who took the same test.

In fact, if *every single person* in the world took that same test, and the average score was 50, then a Z-score of 2.5 would represent an individual who scored 65 on the test.