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 two examples:

Professional mathematicians use highly complex algorithms to find and analyse all these patterns, but for now, let’s start with something much simpler.

Simple Sequences

In mathematics, a sequence is a chain of numbers (or other objects) that usually follow a particular pattern. The individual elements in a sequence are called terms.

Here are a few examples of sequences. Can you find their patterns and calculate 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:

x1, x2, x3, x4, x5, x6, x7, …

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 nth term in the sequence by xnxix2.

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 = xn−1 + n

To get the n-th triangle number, we take the previousfirstnext 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 first termlast termsecond term 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: a formula that tells you the nth term directly, without first having to calculate all the previous ones:

xn =

This is called an explicit formula. 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:

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 rotatedreflecteddilated.

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