Introduction
A sequence, in mathematics, is a string of objects, like numbers, that follow a particular pattern. The individual elements in a sequence are called terms. Some of the simplest sequences can be found in multiplication tables:
- 3, 6, 9, 12, 15, 18, 21, …
Pattern: “add 3 to the previous number to get the next number” - 0, 12, 24, 36, 48, 60, 72, …
Pattern: “add 12 to the previous number to get the next number”
Of course we can come up with much more complicated sequences:
- 10,–2 8,×2 16,–2 14,×2 28,–2 26,×2 52, …
Pattern: “alternatingly subtract 2 and multiply by 2 to get the next number” - 0,+2 2,+4 6,+6 12,+8 20,+10 30,+12 42, …
Pattern: “add increasing even numbers to get the next number”
We can also create sequences based on geometric objects:
Triangle Numbers Pattern: “add increasing integers to get the next number” | |||||
1 | 3 | 6 | 10 | 15 |
Square Numbers Pattern: “add increasing odd numbers to get the next number”
| |||||
1 | 4 | 9 | 16 | 25 |
Note that the sequences of triangle and square numbers also have numerical patterns like the ones we saw at the beginning. To find the following triangle numbers we have to add increasing integers to the last term of the sequence (+2, +3, +4, …). To find the following square numbers we have to add increasing odd numbers (+3, +5, +7, …).
You can also create photography sequences. This is called Action Sequence Photography.
Patterns and Equations
There are different ways in which we can describe the pattern underlying a sequence mathematically. First let us represent every term in a sequence by a variable: let us call the nth term in the sequence x_{n}. The first term of the sequence is represented by x_{1}, the second term by x_{2}, and so on. Thex’s are simply place holders until we have calculated their actual value.
Let us think about the sequence of multiples of 3 above: 3, 6, 9, 12, 15, and so on. If we know any term in the sequence, we can work out the next term by adding 3. With the notation above, x_{n} = x_{n–1} + 3. Every term x_{n} is the previous one, x_{n–1} plus 3. If we know the first term, x_{1} = 3, we work out the following terms step by step:
x_{2}= x_{1}+ 3 = 3 + 3 = 6
x_{3}= x_{2}+ 3 = 6 + 3 = 9
…
An equation that expresses x_{n} in terms of previous values is called a recurrence relation, and these are very useful for calculating the terms of the sequence step by step. However if we were only interested in the 100th term of the sequence, we would have to calculate all terms up to 100. It would be much easier if we had an equation that tells us any term of the sequence, without calculating all the previous ones.
Let us think again about the multiples of 3. Above we found the recurrence relation x_{n} = x_{n–1} + 3. But it is clear that the nth term of the sequence has to be 3 × n. For example, the 4th term is 3 × 4 = 12. An expression for x_{n} = 3n which only depends on n and not any other x’s is called a closed form solution. It has the advantage that we can quickly find very large terms of the sequence without having to calculate all previous terms. The 55th term, for example, is 3 × 55 = 165.
Here is one more interesting sequence:
2, 4, 8, 16, 32, 64, …
The pattern is very simple: we multiply terms by two to get the next term. In this case there is also a closed form solution. By the time we are at term n, we have multiplied the initial 2 by 2 n – 1 times. Therefore x_{n} = 2^{n}. The sequence is therefore called the powers of 2. (Often we start with 2^{0} = 1.)
Find the pattern and continue the following sequences. For the first four sequences also try to find a closed form expression.
- 5, 8, 11, 14, 17, ___, ___, ___, …
- 25, 21, 17, 13, 9, ___, ___, ___, …
- 4, 6, 9, 13, 18, ___, ___, ___, ___, …
- 1, 3, 9, 27, 81, ___, ___, ___, ___, …
- 3, 6, 5, 10, 9, 18, ___, ___, ___, ___, …
- 2, 4, 8, 10, 20, 22, 44, ___, ___, ___, ___, …
- 1, 2, 4, 5, 8, 9, 13, ___, ___, ___, ___, …
- 3, 3, 4, 8, 10, 30, 33, ___, ___, ___, ___, …
- 1, 2, 3, 6, 11, 20, 37, ___, ___, ___, …
Here are the patterns for the sequences. It is not always possible to find simple closed form expressions.
- 5,+3 8,+3 11,+3 14,+3 17,+3 20,+3 23,+3 26, … x_{n} = 2 + 3n
- 25,–4 21,–4 17,–4 13,–4 9,–4 5,–4 1,–4–3, … x_{n} = 29 – 4n
- 4,+2 6,+3 9,+4 13,+5 18,+6 24,+7 31,+8 39,+9 48, … x_{n} = (n^{2} + n)/2 + 3 – this closed form was rather hard to find…
- 1,×3 3,×3 9,×3 27,×3 81,×3 243,×3 729,×3 2187,×3 6561, … x_{n} = 3^{n–1} – the powers of 3
- 3,×2 6,–1 5,×2 10,–1 9,×2 18,–1 17,×2 34,–1 33,×2 66, …
- 2,+2 4,×2 8,+2 10,×2 20,+2 22,×2 44,+2 46,×2 92,+2 94,×2 188, …
- 1,+1 2,+2 4,+1 5,+3 8,+1 9,+4 13,+1 14,+5 19,+1 20,+6 26, …
- 3,×1 3,+1 4,×2 8,+2 10,×3 30,+3 33,×4 132,+4 136,×5 680,+5 685, …
- 1, 2, 3,1+2+3= 6,2+3+6= 11,3+6+11= 20,6+11+20= 37,11+20+37= 68,20+37+68= 125,37+68+125= 230, …
Perfect Numbers
If you have read the article on Prime Numbers, you will know that a divisor of a number n is another number which divides n without remainder. Usually we want to multiply these divisors to get the original number, but let us think about what happens if we instead add them:
number | divisors | sum of divisors | |||||
6 | 1 | 2 | 3 | 6 | perfect number | ||
15 | 1 | 3 | 5 | 9 | deficient number | ||
20 | 1 | 2 | 4 | 5 | 10 | 22 | abundant number |
If the sum of the divisors is bigger than the number itself, we call it an abundant number (relatively many divisors). If the sum of the divisors is smaller than the number itself, we call it a deficient number (relatively few divisors). If the sum of the divisors is equal to the number itself, we call it a perfect number.
Perfect numbers a extremely rare: the smallest six are 6, 28, 496, 8128, 33550336 and 8589869056. All these perfect numbers are even – nobody knows whether there are any odd perfect numbers, but we know that there aren’t any below 10^{1500} (that’s a 1 with 1500 zeros)!
Fibonacci Numbers and the Golden Ratio
One of the most famous sequences is the Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci. We start with 1, 1, … and every new number is the sum of the two previous numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
The recurrence relation for the Fibonacci numbers is x_{n} = x_{n–1} + x_{n–2}.
As before, there is a geometric representation of this sequence. We start with two squares of size 1. Along one edge, we add a new square of size 2, and then another square of size 3. We keep adding squares to the longest edge of the rectangle as shown below. Since the edge of every new square is the sum of the edges of the two previous squares, we get the Fibonacci numbers. If we trace a curve along the corners of these squares, we can make a spiral. This spiral approximates the Golden Spiral. Many similar logarithmic spirals appear in nature, for example Nautilus shells.
While adding more squares, the proportions of the rectangle become closer to a very special shape: the Golden rectangle. The ratio of the sides of the golden rectangle is called the Golden ratio. It is the limit of the ratio of consecutive Fibonacci Numbers.
11 = 1, 21 = 2, 32 = 1.5, 53 = 1.67, 85 = 1.6, 138 = 1.63, …
You can see that these ratios get closer and closer to a particular number around 1.6. This is the Golden ratio and its actual value is 1.61803… Some believe that this ratio is particularly pleasing to humans, and that it underlies the proportions of many buildings, animals or plants.
The Fibonacci numbers have become extremely popular in recreational mathematics, since they seem to appear in many different places in nature.
COMING SOON
COMING SOON
COMING SOON
It is important to remember that nature doesn’t know about Fibonacci numbers or the Golden ratio. Plants and animal populations always grow in the most efficient way, and in some cases it results in these regular patterns.
Pascal’s Triangle
Pascal’s Triangle is an infinite symmetric number pyramid. It has been known since the 10th century to Indian and Arabic mathematicians and the French mathematician Blaise Pascal (1623 – 1662) explored it in more detail and discovered its relation to combinatorics.
In many ways we can think of it as a two-dimensional sequence: we start with a single 1 at the top, and every number in one of the following rows is the sum of the two numbers above. Here is what the first 15 rows look like:
1 | |||||||||||||||||||||||||||||
1 | 1 | ||||||||||||||||||||||||||||
1 | 2 | 1 | |||||||||||||||||||||||||||
1 | 3 | 3 | 1 | ||||||||||||||||||||||||||
1 | 4 | 6 | 4 | 1 | |||||||||||||||||||||||||
1 | 5 | 10 | 10 | 5 | 1 | ||||||||||||||||||||||||
1 | 6 | 15 | 20 | 15 | 6 | 1 | |||||||||||||||||||||||
1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | ||||||||||||||||||||||
1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 | |||||||||||||||||||||
1 | 9 | 36 | 84 | 126 | 126 | 84 | 36 | 9 | 1 | ||||||||||||||||||||
1 | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | 1 | |||||||||||||||||||
1 | 11 | 55 | 165 | 330 | 462 | 462 | 330 | 165 | 55 | 11 | 1 | ||||||||||||||||||
1 | 12 | 66 | 220 | 495 | 792 | 924 | 792 | 495 | 220 | 66 | 12 | 1 | |||||||||||||||||
1 | 13 | 78 | 286 | 715 | 1287 | 1716 | 1716 | 1287 | 715 | 286 | 78 | 13 | 1 | ||||||||||||||||
1 | 14 | 91 | 364 | 1001 | 2002 | 3003 | 3432 | 3003 | 2002 | 1001 | 364 | 91 | 14 | 1 |
Note that the numbers along the edges are always 1’s, but the numbers towards the centre increase very quickly. Like so many other examples before, Pascal’s Triangle is a very complex and interesting object which can be created using a simple pattern. You can find many different sequences in the rows and diagonals of the triangle, and here are a few examples:
Remember that Pascal’s Triangle was created using nothing but addition. Therefore it is surprising that we can find patterns like the powers of 2 or the multiples of Prime numbers, which are intrinsically linked to multiplication. Of course all these patterns occur because of particular mathematical reasons, and it is not hard to prove them.
Another thing we can try is to highlight all cells in the triangle which are divisible by a certain number.
If we were to do the same for even bigger numbers, we would find more and more triangles of increasing size. The patterns are very similar to the Sierpinski Gasket fractal.
As we calculate more rows in Pascal’s triangle, two simple questions arise: which numbers appear in the triangle, and how often does every number appear?
The first question is easily answered: 1 appears infinitely often and every other whole number will eventually appear in the second diagonal on either side. Some numbers also appear in the middle, either twice on either side or once in the centre. These numbers now appear either three or four times in total.
A few numbers appear five or more times. For example, we can see 120 four times in the triangle above (rows 10 and 16) and we know that it will also appear twice in row 120. This makes six occurences in total.
Another example is 3003: it appears twice in rows 14 and 15 each and it will also appear twice in row 3003. But 3003 appears two more times, in the third diagonal on either side of the triangle. Remember that these diagonals are the triangle numbers, and that all triangle numbers are of the form n (n + 1) / 2. We know that 77 × 78 / 2 = 3003, so 3003 is a triangle number. In total, we have found eight occurences of 3003 in the triangle.
We don’t know any other number which appears eigth times in Pascal’s triangle, or whether any number appears more than eigth times. The Singmaster conjecture, named after the American mathematician David Singmaster (*1939), postulates that there is a fixed limit on how often numbers can appear in Pascal’s triangle – but this limit might be larger than eight.
Convergence of Sequences
In some sequences, such as Prime numbers or Perfect numbers, the individual terms are very special and interesting. In other sequences we may only be interested in what happens to the terms as we calculate more and more of them (what happens to x_{n} as n gets very large). Here are a few examples of what could happen (the numbers, for clarity, are represented by dots):
This sequence gets closer and closer to a particular number. We say that it converges. | This sequence doesn’t converge, since it doesn’t keep getting closer to one single number. | This sequence keeps on growing. We say that it diverges. |
Convergence means that the terms keep getting closer to a particular number, and divergence means that the terms keep getting bigger, whether towards infinity or negative infinity. Remember that the sequence of ratios of consecutive Fibonacci numbers above converged to the golden ratio.
Unfortunately “getting closer” is not a particularly precise description in mathematics. A sequence could for example first get very big and then turn around and converge. We don’t really care about what happens at the beginning, only what happens to the most distant terms. All of the following sequences converge:
Here is how mathematicians define the notion of convergence precisely, and this is one of the most important definitions in all of mathematics:
A sequence with terms x_{1}, x_{2}, x_{3}, … tends to a limit y if we can think of any tiny positive number, let us call it ε (the Greek letter Epsilon), and if eventually all terms of the sequence will be within ε of the limit y. This means that there is some (sometimes very big) integer N so that x_{N}, x_{N+1}, x_{N+2}, … are all between y – ε and y + ε.
Using special mathematical notation, it is possible to express this definition without any words. We use ∀ meaning “for all”, ∃ meaning “there exists” and : meaning “such that”:
∀ ε ∃ N : |x_{n} – y| < ε ∀ n > N
For all ε there exists a number N such that the distance |x_{n} – y| between x_{n} and y is less than ε for all n > N.
Sequences and their convergence is studied in an area of mathematics called Analysis. We use sequences to define crucial concepts in mathematics such as series, continuity and differentiation.
Here are a couple of sequences. Which ones converge? Which limit do they converge to?
- 1, 2, 3, 4, 5, 6, …
- 2, 2, 2, 2, 2, 2…
- 1, 1/2, 1/3, 1/4, 1/5, 1/6, …
- 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, …
- 0.1, 0.02, 0.003, 0.0004, 0.00005, 0.00006, …
Here are a couple of sequences. Which ones converge? Which limit do they converge to?
- 1, 2, 3, 4, 5, 6, … keeps getting bigger and hence diverges.
- 2, 2, 2, 2, 2, 2… converges to 2. The distance between all terms and the limit 2 is always 0!
- 1, 1/2, 1/3, 1/4, 1/5, 1/6, … converges to 0. This is why we sometimes say that 1/∞ = 0.
- 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, … neither converges nor diverges.
- 0.1, 0.02, 0.003, 0.0004, 0.00005, 0.00006, … also converges to 0.