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Sequences and PatternsIntroduction

Thời gian đọc: ~25 min

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

Geologists around the world want to predict earthquakes and volcano eruptions. They can try to find patterns in historical data of from seismographs, of the atmosphere, or even animal behaviour. One earthquake, for example, might trigger aftershocks later.

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 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 .

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: 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:

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: