Sequences and PatternsFigurate Numbers
The name for
However, there are many other sequences that are based on certain geometric shapes – some of which you already saw in the introduction. These sequences are often called
Triangle Numbers
The triangle numbers are generated by creating triangles of progressively larger size:
1
3
6
10
15
21
You’ve already seen the recursive formula for triangle numbers:
It is no coincidence that there are always 10 pins when bowling or 15 balls when playing billiards: they are both triangle numbers!
Unfortunately, the recursive formula is not very helpful if we want to find the 100th or 5000th triangle number, without first calculating all the previous ones. But, like we did with arithmetic and geometric sequences, we can try to find an explicit formula for the triangle numbers.
Let’s start with a triangle of size
First, we make a second copy of the triangle, flipped upside down.
Now we can rearrange the two triangles, so that they fit together perfectly, without any gaps or overlaps.
The two triangles now form a rectangle with ${n} rows and ${n + 1} columns – ${n} × ${n + 1} = ${n×(n+1)} circles in total.
The rectangle is exactly twice as large as the original triangle. Therefore the ${n}th triangle number must be half of that: ${n×(n+1)/2}. In general, the nth triangle number is
Triangle numbers seem to pop up everywhere in mathematics, and you’ll see them again throughout this course. One particularly interesting fact is that any whole number can be written as the sum of at most three triangle numbers:
=
+
+
The fact that this works for all whole numbers was first proven in 1796 by the German mathematician
Problem Solving
What is the sum of the first 100 positive
Rather than manually adding up everything, can you use the
Square and Polygonal Numbers
Another sequence that is based on geometric shapes are the square numbers:
1, 4 +3, 9 +5, 16 +7,
You can calculate the numbers is this sequence by squaring every whole number (
The reason for this pattern becomes apparent if we actually draw a square. Every step adds one row and one column. The size of these “corners” starts at 1 and increases by 2 at every step – thereby forming the sequence of odd numbers.
This also means that the nth square number is just the sum of the first n odd numbers! For example, the sum of the first 6 odd numbers is
In addition, every square number is also the sum of two consecutive
After triangle and square numbers, we can keep on going with larger
For example, if we use polygons with
Can you find recursive and explicit formulas for the nth polygonal number that has k sides? And do you notice any other interesting patterns for larger polygons?
Tetrahedral and Cubic Numbers
Of course, we also don’t have to limit ourselves to two-dimensional shapes and patterns. We could stack spheres to form small pyramids, just like how you would stack oranges in a supermarket:
1
20
35
Mathematicians often call these pyramids
Just like we did for triangle and square numbers, let’s look at the differences between consecutive tetrahedral numbers:
1, 4 +3, 10 +6, 20 +10, 35 +15,
The differences are the triangle numbers: 3, 6, 10, 15, 21, 28, …! To get the next tetrahedral number, you simply add the next
For example, the 5th tetrahedral number is 20 +
We can also find an
For example, the 5th tetrahedral number is 5 × 6 × 7 ÷ 6 =
Tetrahedral numbers appear wherever objects get stacked into pyramids: oranges at a market stall, cannonballs in an old fort, or gifts in the song “The 12 Days of Christmas” – which we will count in a moment. Here is the pyramid with 6 layers, containing
Cubic Numbers
There is one more sequence hiding inside these pyramids. If we stack square layers on top of each other – or simply multiply a number by itself twice – we get the cubic numbers:
1, 8 +7, 27 +19, 64 +37, 125 +61,
Unlike arithmetic or geometric sequences, the cubic numbers have no constant difference or ratio: the differences (7, 19, 37, 61, 91, …) keep changing, and even the differences between those differences (12, 18, 24, …) keep growing!
But there is a very simple
Cubic numbers hide one of the most beautiful identities in all of mathematics. Move the slider to arrange the cubes
With n = 1: 1
= (1 + 2 + … + n)² = 1² = 1
The coloured corners are the cubes
For example,
The 12 Days of Christmas
Tetrahedral numbers have one famous real-world appearance: the song “The 12 Days of Christmas”. On day 1, your true love gives you 1 gift. On day 2, you receive 2 turtle doves plus another partridge – that’s 3 gifts. On day 3 you get 3 + 2 + 1 = 6 gifts, and so on: the number of gifts on each day is a
| Day | Gifts that day | Total gifts |
| 1 | 1 | 1 |
| 2 | 3 | 4 |
| 3 | 10 | |
| 4 | 10 | |
| 5 | 15 | 35 |
| 6 | 56 | |
| 7 | 28 | 84 |
| 8 | 36 | |
| 9 | 45 | 165 |
| 10 | 55 | 220 |
| 11 | 286 | |
| 12 | 78 |
By the end of the song you have received 364 gifts in total:
Problem Solving
1. How many gifts would you receive in “The n Days of Christmas”, for any value of n? Which formula do you get?
2. The running totals satisfy