Geometry Pattern Formula
Hello All, thank you for your help.
I am at a complete loss in this problem:
Attatched are the triangular numbers used to do the problem, and a table clarifying a few things.
The "n" row on the table states what n equals.
The "nth Triangular Number" row states how many dots there are when n=a certain number. (So when n=1, there is one dot.)
I am to find a formula for the nth triangular number.
The book says that each triangular number could be thought of as half the area of a rectangle whose width is the same as the triangle number, and it's height is n+1. So if n=2, the box is thought of as 2 units wide and 3 units tall, for a total of 6 units.
It also says that the formula for this problem is:
I have no idea how one would arrive at such an explanation for a pattern of said triangular numbers, nor how one would arrive at such a formula.
Any help on the correct way to solve a formula such as this and how to find the formula would be most appreciated.
Thanks a lot,
sequence of triangular numbers ...
1, 3, 6, 10, 15, ...
double them ...
2, 6, 12, 20, 30, ...
factor each term ...
1*2, 2*3, 3*4, 4*5, 5*6, ... , n(n+1)
undouble (halve) ...
1, 3, 6, 10, 15, ... , n(n+1)/2
That hint goes like . . .
Suppose we want to know without counting them.
We have: .
Left-justify the dots: .
Add a mirror-image copy of the triangle: .
We have a 5-by-6 rectangle with a total of: . dots.
The triangle contains half that many dots.
In general: .