# Sequences and PatternsIntroduction

Many professions that use mathematics are interested in one specific aspect – *finding patterns*, and being able to predict the future. Here are a few examples:

In the last decade, **police departments** around the world have started to rely more on mathematics. Special algorithms can use the data of past crimes to predict when and where crimes might occur in the future. For example, the *PredPol* system (short for “predictive policing”), helped decrease the crime rate in parts of Los Angeles by 12%!

It turns out that **earthquakes** follow similar patterns to crimes. Just like one crime might trigger retaliations, an earthquake might trigger aftershocks. In mathematics, this is called a “self-exciting process”, and there are equations that help predict when the next one might happen.

Bankers also look at historical data of stock prices, interest rates and currency exchange rates to estimate how **financial markets** might change in the future. Being able to predict if the value of a stock will go up or down can be extremely lucrative!

Professional mathematicians use highly complex algorithms to find and analyse all these patterns, but we are going to start with something a bit more basic.

## Simple Sequences

In mathematics, a **sequence****terms**

Here are a few examples of sequences. Can you find their patterns and calculate the next two terms?

*3*, *6+3*, *9+3*, *12+3*, *15+3*, *+3*, … Pattern: “Add 3 to the previous number to get the next one.”

*4*, *10+6*, *16+6*, *22+6*, *28+6*, *+6*,

*, … Pattern: “Add 6 to the previous number to get the next one.”* +6

*3*, *4+1*, *7+3*, *8+1*, *11+3*, *+1*,

*, … Pattern: “Alternatingly add 1 and add 3 to the previous number, to get the next one.”* +3

*1*, *2×2*, *4×2*, *8×2*, *16×2*, *×2*,

*, … Pattern: “Multiply the previous number by 2, to get the next one.”* ×2

The dots (…) at the end simply mean that the sequence can go on forever. When referring to sequences like this in mathematics, we often represent every term by a special

The small number after the *x* is called a **subscript**, and indicates the position of the term in the sequence. This means that we can represent the *n*th term in the sequence by

## Triangle and Square Numbers

Sequences in mathematics don’t always have to be numbers. Here is a sequence that consists of geometric shapes – triangles of increasing size:

**1**

**3**

**6**

At every step, we’re adding one more row to the previous triangle. The length of these new rows also increases by one every time. Can you see the pattern?

*1*, *3+2*, *6+3*, *10+4*, *15+5*, *21+6* *+7*,

*, …* +8

We can also describe this pattern using a special

*n*

To get the *n*-th triangle number, we take the *n*. For example, if *n* =

A formula that expresses **recursive formula**

Another sequence which consists of geometric shapes are the **square numbers**. Every term is formed by increasingly large squares:

**1**

**4**

**9**

For the triangle numbers we found a recursive formula that tells you the *next* term of the sequence as a function of of its *previous* terms. For square numbers we can do even better: a formula that tells you the *n*th term directly, without first having to calculate all the previous ones:

=

This is called an **explicit formula**

Let’s summarise all the definitions we have seen so far:

A **sequence****terms**

A **recursive formula***n*th term as a function of

An **explicit formula***n*th term as a function of

## Action Sequence Photography

In the following sections you will learn about many different mathematical sequences, surprising patterns, and unexpected applications.

First, though, let’s look at something completely different: **action sequence photography**. A photographer takes many shots in quick succession, and then merges them into a single image:

Can you see how the skier forms a sequence? The pattern is not addition or multiplication, but a geometric

Here are a few more examples of action sequence photography for your enjoyment: