Here is a pattern of numbers you might recognise:
. 1 3 6 10 15 21 28 36 45 55 66 78 91 …
These are triangle numbers. Three dots makes triangle; add a row of three dots and keep the triangle, add four dots and you (could) still have a triangle. Hence, obviously, these are called the triangle numbers. Finding a formula for these is covered on an adjacent page, but one way to do that is to write (spot) factors of the numbers. I use dot for ‘multiply’:
6=2.3; 10=2.5; 15=3.5; 21=3.7; 28=4.7; 36=4.9, 45=5.9, 55=5.11, 66=6.11
If I write those slightly differently, as
3.2 2.5 5.3 3.7 7.4 4.9 9.5 5.11, then there is an apparent repetition in the adjacent figures—looking across the gaps—showing one series 2,3,4,5 and another series, 3,5,7,9. You may need to look hard to see this.
If I try to write a formula based on connecting the tenth term, 55, with the ten, I see half of ten times eleven, or n/2. (n+1). If I try to apply this to the ninth term, 45=5.9 I can see that this, n. (n+1)/2, also works and the formula has been found.
Writing successive differences, as explained in this sheet entitled Formulae, shows that the first line with a constant difference tells you the order of the formula.
. 1 1 1 1 1 1 1 1 1 1 1 …. second differences constant
. 2 3 4 5 6 7 8 9 10 11 12 13 … first differences
. 1 3 6 10 15 21 28 36 45 55 66 78 91 …
In this case, the second difference is one [the first differences are 1,2,3,4,5…] and that figure 1 tells you that the formula will begin with n²/2. To test this idea, write the squares out, see that their first differences are the odd numbers and therefore the second differences are all two. Adding some constant to the squares doesn’t affect the first difference and adding something xn to each number doesn’t affect the second difference, so this stands inspection.
Exercise:
1 Write down the first ten squares, write their differences and the second differences and see that these are all two.
2 Explore this pattern by writing differences: 1 12 25 40 57 75 96
3 Explore this pattern: 4 13 24 37 52 69 87 108. It is (n+4)² – 12.
4 Similarly 104 113 124 137 152 169 187 208 This is Q3 + 100
5 Find a way of drawing the six-dot triangle and the ten-dot triangle to make a 16-dot square. Fit together two adjacent triangle numbers (as dots) to make a square. Express the dimensions of the rectangle in terms of the n of the smaller number.
We can make the whole pattern more complicated, like these, called pyramid numbers:
. 1 4 10 20 35 56
I have left you space to write in the differences. The differences here are themselves the triangle numbers. That in turn means that the third difference is constant (and one).
Exercise:
5 Write down the differences for: 1 8 27 64 125 216 343.
Keep writing differences until you have a row of sixes. Since I gave you a set of cubes, this tells you that any formula whose biggest term is a cube will have a constant difference of six. A series or sequence with a third constant difference of 12 will have a formula beginning with 2n³.
6 Write differences for 51 58 77 114 175 266 and spot the formula that connects them.
7 Write differences for 0 6 24 60 120 210 336 and spot the formula that connects them.
8 Write differences for 1 1 2 3 5 8 13 21 34 A formula is extremely difficult to write, but the relationship between any number and its predecessors can be seen. Find the next two numbers in this series which are also triangle numbers.
9 Given that the constant difference for squares is two and the constant difference for cubes is six, what do you think the constant difference for quartics (x⁴) will be? Why?
Test your idea by writing out differences for 1 16 81 256 625 1296 2401 4096. Now correct your theory if necessary and, by writing the series 2 6 (x⁴) predict what the next number will be, and why. You may be sufficiently convinced to merely confirm that the series 1 32 243 1024 3025 7776 16807 32768 behaves as you expect.
An awful lot of real maths relies upon you being able to spot patterns; real in the sense that real people outside education are spending a lot of time looking for patterns and being paid a lot of money to do so. How do you think people make money from the stock market? Answer; by predicting what the short-term future will hold and, in effect, betting upon their predictions. Getting this right makes you (and your clients) rich. So there is a lot of benefit to be found in studying trends and patterns. Humans are quite good at seeing patterns and mathematicians are better than most at this. We do a lot of business in moving money around and looking for patterns in spending is what many people are doing. Increasingly, we are writing software to do this hunting, so the problem moves to become one of finding good data, sometimes (and not always accurately) called data mining. I suspect that the future will cause us to concentrate on finding ways of ascertaining the value/usability/reliability of available data, once we are sure that we have tools (computer software) good at finding any patterns. Interpretation will remain a problem. Determining the reliability of data is something we explore in statistics.
DJS 20130429&30
I note, in 2017, that 'data mining’ has become a relatively common term, where it was a term I’d come across in the early 90s when writing about a specialist area within computing. Since writing this page I have found many more references fulfilling the predictions of this last paragraph. Prehaps this is an effect of my students moving into prediction? I hope they did — and made lots of money by being more right than their neighbours. Or less wrong.
I I note, in 2017
