We’re s--o excited about this!!! As in, E–X–C–I–T–E–D!!! Indeed, this is expected, considering what is to be expounded: “any” geometric sequence can be encapsulated as slopes of a simple triangle like the one below:

And the procedure is easy: first find a triangle with slopes , then the remaining terms can be obtained as slopes of line segments drawn from a fixed vertex to a fixed side .

## Positive common ratio

that is, (positive) powers of .

#### Example

Find coordinates for the vertices of with slopes .

There are different ways to obtain a triangle whose slopes are . One can begin with an arbitrary point . Then the two other points can be chosen as and . With these, the slopes of sides will be , respectively.

In the present case, let’s first choose as , for simplicity. Since the sequence is , we have and . So and .

The entire geometric sequence can be obtained from the above triangle.

#### Example

For each , PROVE that lies __within__ , where is the point and is the point .

Note that is a point on . Let’s use distances to show that it’s indeed internal. We prove that .

This proves that for each , the point lies internally on .

#### Example

For each , let . PROVE that the slope of is , where is the point .

This follows by direct calculation:

Thus, when , the corresponding slopes will be ; that is, .

By defining and using with vertices at , we’ve seen that the entire geometric progression can be obtained as slopes of the line segments , together with with slope . Notice that when , becomes and when , becomes . What happens to when ?

as expected.

Put , then has coordinates . Together with , the slope of line segment is

as expected.

Put , then has coordinates . Together with , the slope of line segment is

as expected.

Three things to note from the above calculation:

- the points are defined relative to the chosen coordinates for ;
- the line segments that describe the geometric progression do not get to the midpoint of side ;
- the roles of vertex and side cannot be altered; doing so will distort the orderly arrangement of the line segments. See the next two examples.

#### Example

Given with vertices , find coordinates for a point on such that the slope of the line segment is .

Let’s find the equations of and and then solve the resulting linear system:

Thus, is the point . It is external to side — we don’t want that.

#### Example

Given with vertices , find coordinates for a point on such that the slope of the line segment is .

Let’s find the equations of and and then solve the resulting linear system:

Thus, is the point . It is external to side — we don’t want that.

#### Example

Find coordinates for the vertices of whose side slopes are .

Take . Then the slope of is , the slope of is , and the slope of is .

#### Example

Let . For each , PROVE that the slope of the line segment is , where is the point .

Observe that each lies __within__ (). By direct calculation:

Thus, we obtain the geometric progression by putting . Together with the first three terms that are the slopes of the original triangle, we obtain the entire geometric progression .

#### Example

Explain how to obtain the entire geometric progression as slopes.

Easy-peasy.

Begin with a in which the slopes of the sides are . To this end, take . Then the slope of is , the slope of is , and the slope of is .

Next, let . Then, by direct calculation (as in and ), the slope of is , for . Therefore, letting successively, we obtain the terms . If we now append the slopes of the sides of (which are ), we obtain the entire geometric progression .

## Reciprocal terms

In addition to obtaining , we also get , once we go past a certain point; let’s call it a terminal point (see the exercises at the end ).

#### Example

Given with vertices , find coordinates for a point on such that the slope of is .

In Example we had . Put , then . Take . Then is an internal point on ; further, the slope of is

as desired. Similarly, by putting , we obtain the terms . Thus, inside a simple, single triangle lies the __infinite__ geometric progression

disguised as slopes.

#### Example

Given with vertices , let . PROVE that .

Put , then . Put , then , as also found in a previous example. By the distance formula:

Observe that the slopes of the sides of are ; they form a geometric progression with a common ratio of , which is the __square__ of the common ratio of the geometric progression formed by the slopes of the sides of the parent .

## Takeaway

Slopes are usually introduced sometime in grade (or earlier), while geometric sequences are usually studied in grade (or earlier). Here, we’ve managed to show a close connection between the two concepts, using the medium of a triangle.

Throughout we worked with positive integer common ratios. But the common ratios can also be negative, and in such cases things become much more exciting.

## Tasks

- (Depressed cubic) Consider with vertices . PROVE that:
- the foot of the altitude from vertex is ;
- the length of the altitude from vertex is ;
- the area of is (this becomes a depressed cubic if written in the form . Moreover, if is an integer ( of course), then this area is always a positive integer).

- Let . PROVE that as .
- Let . PROVE that as .
- (Special points) Given with vertices , PROVE that there are (internal) points and on such that:
- the slope of is
__undefined__and the slope of is__zero__( for vertical, for horizontal); - and trisect side ();
- the areas of are sq. unit each (making the overall area of to be sq. units).

- the slope of is
- (Higher powers) Given with vertices at , the slopes of its sides form a geometric progression with common ratio . Find two points and on such that the slopes of the sides of form a geometric progression with common ratio .
- (Negative region) Given with vertices , find two points and on such that the slope of any line segment , where is a point between and , is negative.
- (Base five) Explain how to generate the geometric progression using slopes.
- (Base six) Explain how to generate the geometric progression using slopes.
- (Base seven) Explain how to generate the geometric progression using slopes.
- (Terminal point) Given with vertices , determine the coordinates of the terminal point on (a point on such that the slope of the line segment is ; this is also where the slopes of the line segments from vertex change from positive powers of to negative powers of ).