Sequences and PatternsSpecial Sequences
In addition to An arithmetic sequence (sometimes called arithmetic progression) is a sequence of numbers in which the difference d between consecutive terms is always constant. A geometric sequence (sometimes called geometric progression) is a sequence of numbers in which the ratio r between consecutive terms is always constant. The sequence of Fibonacci numbers starts with 1, 1. Every following term is the sum of the two previous terms, which means that the recursive formula is Figurate numbers are numbers that can be represented using geometric shapes. Examples include square, triangle and tetrahedral numbers.
Prime Numbers
One example that you’ve already seen before are the A prime number is a positive integer that has no factors other than 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, … A number a is a factor (or divisor) of a number b, if you can divide b by a without remainder.
Here are the first few prime numbers:
2, 3, 5, 7, 11,
Unfortunately, prime numbers don’t follow a simple pattern or recursive formula. Sometimes they appear directly next to each other (these are called Twin primes are pairs of prime numbers like 17 and 19 or 101 and 103, which are exactly two apart. It is unknown if there are infinitely many pairs of twin primes.
Prime numbers also don’t have a simple geometric representation like A triangle number is an integer that can be represented as an equilateral triangle of dots. The formula for the nth triangle number is A square number is a number that can be expressed as the square of another integer. The first square numbers are 1, 4, 9, 16, 25, …
You can learn more about these and other properties of prime numbers in our course on Divisibility and Primes. They are some of the most important and most mysterious concepts in mathematics!
Perfect Numbers
To determine if a number is A prime number is a positive integer that has no factors other than 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, … A number a is a factor (or divisor) of a number b, if you can divide b by a without remainder.
Number | Factors | Sum of Factors |
5 | 1 | 1 |
6 | 1 2 3 | 6 |
7 | 1 | 1 |
8 | 1 2 4 | 7 |
9 | 1 3 | 4 |
10 | 1 2 5 | |
11 | 1 | |
12 | 1 2 3 4 6 | |
13 | 1 | |
14 | 1 2 7 | |
15 | 1 3 5 | |
16 | 1 2 4 8 | |
17 | 1 | |
18 | 1 2 3 6 9 |
Let’s compare these numbers with their sum of factors:
For most numbers, the sum of its factors is
For a few numbers, the sum of its factors is greater than itself. These numbers are called abundant numbers.
Only one number in the list above has a sum of factors that is equal to itself: A perfect number is a number that is equal to the sum of its divisors (excluding itself). For example,
The next perfect number is 28, because if we add up all its factors we get
6, 28, 496, 8,128, 33,550,336, 8,589,869,056, 137,438,691,328, 2,305,843,008,139,952,128, …
Notice that all of these numbers are
Perfect numbers were first studied by ancient Greek mathematicians like Euclid of Alexandria (Εὐκλείδης, around 300 BCE) was a Greek mathematician and is often called the father of geometry. His book The Elements first introduced Euclidean geometry, defines its five axioms, and contains many important proofs in geometry and number theory – including that there are infinitely many prime numbers. It is one of the most influential books ever published, and was used as textbook in mathematics until the 19th century. Euclid taught mathematics in Alexandria, but not much else is known about his life. Pythagoras of Samos (c. 570 – 495 BCE) was a Greek philosopher and mathematician. He is best known for proving Pythagoras’ Theorem, but made many other mathematical and scientific discoveries. Pythagoras tried to explain music in a mathematical way, and discovered that two tones sound “nice” together (consonant) if the ratio of their frequencies is a simple fraction. He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules – but the school was eventually burned down by their adversaries. Nicomachus of Gerasa (c. 60 – 120) was an ancient Greek mathematician who also spent much time thinking about the mystical properties of numbers. His book Introduction to Arithmetic contains the first mention of perfect numbers.
Today, mathematicians have used computers to check the first 101500 numbers (that’s a 1 followed by 1500 zeros), but without success: all perfect numbers they found were even. To this day, it is still unknown whether there are any odd perfect numbers, making it the oldest unsolved problem in all of mathematics!
The Hailstone Sequence
Most of the sequences we have seen so far had a single rule or pattern. But there is no reason why we can’t combine multiple different ones – for example a recursive formula like this:
If | |
If |
Let’s start with
5,
It looks like after a few terms, the sequence reaches a “cycle”: 4, 2, 1 will continue to repeat over and over again, forever.
Of course, we could have picked a different starting point, like
, 4, 2, 1, 4, 2, 1, 4, 2, 1, …
It seems like the length of the sequence varies a lot, but it will always end up in a 4, 2, 1 cycle – no matter what first number we pick. We can even visualise the terms of the sequence in a chart:
Notice how some starting points end very quickly, while others (like or ) take more than one hundreds steps before they reach the 4, 2, 1 cycle.
All sequences that follow this recursive formula are called Hailstone sequences can be generated by picking any integer as starting point, and then following these rules:
In 1937, the mathematician Lothar Collatz (1910 – 1990) was a German mathematician working on differential equations and optimisation problems. The Collatz conjecture or
However, there are infinitely many integers. It is impossible to check each of them, and no one has been able to find a A proof is a logical argument that shows beyond any doubt that a certain statement is true.
Just like the search for odd perfect numbers, this is still an open problem in mathematics. It is amazing that these simple patterns for sequences can lead to questions that have mystified even the best mathematicians in the world for centuries!
The Look-and-Say Sequence
Here is one more sequence that is a bit different from all the ones you’ve seen above. Can you find the pattern?
1, 11, 21, 1211, 111221, 312211, …
This sequence is called the Look-and-Say sequence, and the pattern is just what the name says: you start with a 1, and every following term is what you get if you “read out loud” the previous one. Here is an example:
Can you now find the next terms?
…, 312211,
This sequence is often used as a puzzle to trip up mathematicians – because the pattern appears to be completely non-mathematical. However, as it turns out, the sequence has many interesting properties. For example, every term ends in
The British mathematician John Horton Conway (1937 – 2020) was a British mathematician who worked at Cambridge and Princeton University. He was a fellow of the Royal Society, and the first recipient of the Pólya Prize. He explored the underlying mathematics of everyday objects like knots and games, and he contributed to group theory, number theory and many other areas of mathematics. Conway is known for inventing “Conway’s Game of Life”, a cellular automaton with fascinating properties.
The Sequence Quiz
You’ve now seen countless different mathematical sequences – some based on geometric shapes, some that follow specific formulas, and others that seem to behave almost randomly.
In this quiz you can combine all your knowledge about sequences. There is just one goal: find the pattern and calculate the next two terms!
Find the next number
7, 11, 15, 19, 23, 27,
11, 14, 18, 23, 29, 36,
3, 7, 6, 10, 9, 13,
2, 4, 6, 12, 14, 28,
1, 1, 2, 3, 5, 8,
27, 28, 30, 15, 16, 18,
1, 9, 25, 49, 81, 121,