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Sequences and PatternsFigurate Numbers

Reading time: ~20 min

The name for geometric sequences is pretty confusing, because they don’t have anything to do with geometry. In fact, the name was developed hundreds of years ago, when mathematicians thought about multiplication and square roots in a much more geometric way.

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 figurate numbers, and in this section we will have a closer look at some of them.

Triangle Numbers

The triangle numbers are generated by creating triangles of progressively larger size:













You’ve already seen the recursive formula for triangle numbers: xn= .

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.

COMING SOON: Animated Proof for the Triangle Number Formula

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 Carl Friedrich Gauss – at the age of 19!

Problem Solving

What is the sum of the first 100 positive integers? In other words, what is the value of


Rather than manually adding up everything, can you use the triangle numbers to help you? What about the sum of the first 1000 positive integers?

Square and Polygonal Numbers

Another sequence that is based on geometric shapes are the square numbers:

1, 4 +3, 9 +5, 16 +7, +9, +11, +13, +15, …

You can calculate the numbers is this sequence by squaring every whole number (12, 22, 32, …), but it turns out that there is another pattern: the differences between consecutive square numbers are the in increasing order!

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

1+3+5+7+9+11= .

1 3 5 7 9 11 13

In addition, every square number is also the sum of two consecutive triangle numbers. For example, ${n×n} = ${n×(n+1)/2} + ${n×(n-1)/2}. Can you see how we can split every square along its diagonal, into two triangles?


After triangle and square numbers, we can keep on going with larger polygons. The resulting number sequences are called polygonal numbers.

For example, if we use polygons with ${k} sides, we get the sequence of ${polygonName(k)} numbers.

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:




Mathematicians often call these pyramids tetrahedra, and the resulting sequence tetrahedral numbers.

COMING SOON: More on Tetrahedral numbers, Cubic numbers, and the 12 days of Christmas.