Triangular Number Sequence
This is the Triangular Number Sequence:
|1, 3, 6, 10, 15, 21, 28, 36, 45, ...|
This sequence comes from a pattern of dots that form a triangle:
By adding another row of dots and counting all the dots we can
find the next number of the sequence.
- The first triangle has just one dot.
- The second triangle has another row with 2 extra dots, making 1 + 2 = 3
- The third triangle has another row with 3 extra dots, making 1 + 2 + 3 = 6
- The fourth has 1 + 2 + 3 + 4 = 10
How may dots in the 60th triangle?
We can make a "Rule" so we can calculate any triangular number.
First, rearrange the dots like this:
Then double the number of dots, and form them into a rectangle:
Now it is easy to work out how many dots: just multiply n by n+1
Dots in rectangle = n(n+1)
But remember we doubled the number of dots, so
Dots in triangle = n(n+1)/2
We can use xn to mean "dots in triangle n", so we get the rule:
Rule: xn = n(n+1)/2
Example: the 5th Triangular Number is
x5 = 5(5+1)/2 = 15
Example: the 60th is
x60 = 60(60+1)/2 = 1830
Wasn't it much easier to use the formula than to add up all those dots?
Example: You are stacking logs.
There is enough ground for you to lay 22 logs side-by-side.
How many logs can you fit in the stack?
x22 = 22(22+1)/2 = 253
You may get tired, and the stack may be dangerously high, but you can fit 253 logs in it!