Math (Reimy) - Reiny, Thursday, June 25, 2009 at 4:46pm
You are welcome
Just as an afterthought...
There is also a set of numbers called the pyramidal numbers, they are ...
1 4 10 20 ....
Can you figure out the pattern?
(Let their name be a hint.)
http://en.wikipedia.org/wiki/Tetrahedral_number
look at the rotating triangular pyramids and count the number of balls for the different layers
top layer: 1 ball
top two layers: 4 balls
top three layers: 10 balls
etc.
you can figure out the number of balls for any number of layers n by
n(n+1)(n+2)/6
e.g. if n=3
3(4)(5)/6 = 10
To figure out the pattern in the pyramidal numbers, let's look at the differences between consecutive terms:
1 - 0 = 1
4 - 1 = 3
10 - 4 = 6
20 - 10 = 10
The next difference would be 15.
If we look closely, we can see that the differences are getting bigger by consecutive odd numbers: 1, 3, 5, 7, 9, and so on.
Now, let's look at those differences: 1, 3, 6, 10, 15.
If we look closely again, we can see that these differences themselves are forming a pattern. The second difference is obtained by summing the consecutive numbers:
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
10 + 5 = 15
The second differences are the same as the triangular numbers: 1, 3, 6, 10, 15.
From this, we can conclude that the pattern in the pyramidal numbers is that each term is the sum of the triangular numbers up to that term.
For example:
1 = 1 (the sum of the first triangular number)
4 = 1 + 3 (the sum of the first two triangular numbers)
10 = 1 + 3 + 6 (the sum of the first three triangular numbers)
20 = 1 + 3 + 6 + 10 (the sum of the first four triangular numbers)
So, the pattern in the pyramidal numbers is that each term is the sum of the triangular numbers up to that term.