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
The triangle numbers are generated by creating triangles of progressively larger size:
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 billiard: 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
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 2-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
COMING SOON: More on Tetrahedral numbers, Cubic numbers, and the 12 days of Christmas.