Thread: 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:
S-1/2(Width)(Length)=1/2(n)(n+1)

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,
Rocker1414

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

Hello, Rocker1414!

Attatched are the triangular numbers used to do the problem, and a table clarifying a few things.

The row on the table states what n equals.
The row states how many dots there are when = a certain number.

I am to find a formula for the triangular number.

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.

Therefore: .


In general: .