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Sequences and Patterns
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Introduction
Arithmetic and Geometric Sequences
Figurate Numbers
Sequences as Functions
Fibonacci Numbers
Special Sequences
Pascal’s Triangle
Limits and Convergence

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Sequences and PatternsSpecial Sequences

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In addition to arithmetic and geometric sequences, Fibonacci numbers and figurate numbers, there are countless interesting sequences that don’t follow a similar, regular pattern.

Prime Numbers

One example that you’ve already seen before are the Prime numbers. We say that a number is prime if it has no factors .

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), and sometimes there are huge gaps between them. They seem to be distributed almost randomly!

Prime numbers also don’t have a simple geometric representation like triangle or square numbers, but with a bit of work we can reveal interesting patterns:

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If we cross out all multiples of small integers, the remaining numbers must all be prime. This method is called the Sieve of Eratosthenes.

If we draw a chart that increases by 1 whenever there is a prime number, we get a “stepped” function with fascinating properties.

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 prime, we have to find all of its factors. Usually we would multiply these factors to get back the original number, but let’s see what happens if we add up all factors of a number (excluding the number itself):

NumberFactorsSum of Factors
5
1
1
6
1
2
3
6
7
1
1
8
1
2
4
7
9
1
3
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10
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11
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15
1
3
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16
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1
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1
2
3
6
9

Let’s compare these numbers with their sum of factors:

For most numbers, the sum of its factors is itself. These numbers are called deficient numbers.

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: . This is called a perfect number.

The next perfect number is 28, because if we add up all its factors we get 1+2+4+7+14=28. After that, perfect numbers become much rarer:

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 . It turns out that they are also all triangle numbers!

Perfect numbers were first studied by ancient Greek mathematicians like Euclid, Pythagoras and Nicomachus, more than 2000 years ago. They calculated the first few perfect numbers, and wondered if there might be any odd ones.

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!

Euclid of Alexandria

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 xn is even:xn+1=xn/2
If xn is odd:xn+1=3xn+1

Let’s start with x1=5 and see what happens:

5, ×3 +1, ÷2, ÷2, ÷2, ÷2, ×3 +1, ÷2, ÷2, …

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 ${n}. Then the sequence would look like this:

, 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:

Start value:${n}

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, because they seem to move randomly up and down before reaching the 4, 2, 1 cycle – just like hailstones that move up and down in a cloud before crashing to Earth.

In 1937, the mathematician Lothar Collatz proposed that every hailstone sequence eventually ends in a 4, 2, 1 cycle – whatever starting value you pick. You’ve already checked a few starting points above, and computers have actually tried all numbers up to 1020 – that’s 100 billion billion or a 1 followed by twenty zeros.

However, there are infinitely many integers. It is impossible to check each of them, and no one has been able to find a proof that works for all.

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 , and no digit larger than ever gets used.

The British mathematician John Conway discovered that, no matter what number you pick as starting value, the sequence will eventually split into distinct “sections” that no longer interact with each other. Conway called this the Cosmological Theorem, and named the different sections using the chemical elements Hydrogen, Helium, Lithium, …, up to Plutonium.

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, , , … Pattern: Always +4

11, 14, 18, 23, 29, 36, , , … Pattern: +3, +4, +5, +6, …

3, 7, 6, 10, 9, 13, , , … Pattern: +4, –1, +4, –1, …

2, 4, 6, 12, 14, 28, , , … Pattern: ×2, +2, ×2, +2, …

1, 1, 2, 3, 5, 8, , , … Pattern: Fibonacci Numbers

27, 28, 30, 15, 16, 18, , , … Pattern: +1, +2, ÷2, +1, +2, ÷2, …

1, 9, 25, 49, 81, 121, , , … Pattern: Odd square numbers

Archie