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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 processes”, and there are equations that help predict when the next one might happen.

Bankers also look at historic 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 is a collection of numbers (or other objects) that follow a particular pattern. The individual elements in a sequence are called terms.

Here are a few examples of sequences. Can you find the pattern and fill in the next two terms?

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

4, 10, 16, 22, 28, , , … Pattern: “Add 6 to the previous number to get the next one.”

3, 4, 7, 8, 11, , , … Pattern: “Alternatingly add 1 and add 3 to the previous number, to get the next one.”

1, 2, 4, 8, 16, , , … Pattern: “Multiply the previous number by 2, to get the next one.”

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 variable:

x1x2x3x4x5x6x7, …

The small number after the x is called a subscript, and indicates the position of the term in the sequence. For example, the nth term in the sequence will be represented by the variable xn.

You might think that it would be easier to label the terms in the sequence as a, b, c, d, and so on. However you’ll eventually , while the sequence might go on forever!

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:










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 formula:

xn = xn1 + n

To get the n-th triangle number, we take the triangle number and add n. For example, if n = ${n}, the formula becomes x${n} = x${n-1} + ${n}.

A formula that expresses xn as a function of previous terms in the sequence is called a recursive formula. As long as you know the in the sequence, you can calculate all the following ones.

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










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: an equation that tells you the nth term directly, without first having to calculate all the previous ones:

xn =

Equations like this are called explicit formulas. We can use it, for example, to calculate that the 13th square number is , without first finding the previous 12 square numbers.

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

A sequence is a list of numbers, geometric shapes or other objects, that follow a specific pattern. The individual items in the sequence are called terms, and represented by variables like xn.

A recursive formula for a sequence tells you the value of the nth term as a function of . You also have to specify the first term(s).

An explicit formula for a sequence tells you the value of the nth term as a function of , without referring to other terms in the sequence.

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 transformation. Between consecutive steps, the skier is both translated and .

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