Project: Sequences of Points, Circles, Line Segments, Polygons -

A sequence of polygons

The objective of this project is to create and analyze sequences using GeoGebra and your knowledge of arithmetic sequences.

Project description: Arithmetic Sequences Project tasks

Explore: Use the following instructions to explore various sequence diagrams.

To begin, open a GeoGebra classic page, (close the keyboard) then open a long input bar at the bottom as follows:

For each command given as an example, either type carefully to include all brackets or copy and paste into the input bar on the GeoGebra page.

A Sequence

A sequence until now has simply been a list of numbers. The sequence command in GeoGebra is as follows:

Fill in the input options as follows:

Sequence(2n+1,n,1,10,1)

Press enter, to generate a simple sequence of numbers:

Sequence of points

A point is an ordered pair: it contains an $x$ and a $y$ coordinate. To draw a graph of an arithmetic sequence, we use the $x$ values $1, 2, 3, \dots$ to represent the place in the sequence.

Copy/Type in the sequence command

Sequence((n,2n+1),n,1,10,1)

This generates the set of points

$(1,3), (2,5), (3,7), (4,9), ... , (10,21)$

Sequences of Circles

The circle command requires a point for the center, and a radius.

Example 1. Clear your GeoGebra screen and type in the following set of circles, one at a time:

Circle((1,1),1)

Circle((2,2),1)

Circle((3,3),1)

Circle((4,4),1)

This should give four circles, all with different centers but radius 1.

What they have in common is that the centers are at the point (n,n) between $n=1$ and $n=4$

Now, clear the screen and enter all of them together with one sequence command:

Sequence(circle((n,n),1),n,1,4,1)

This means, circle center (n,n) with radius 1, for all values of n between 1 and 4, going up in ones.

Example 2.

Now enter

Sequence(circle((n,n),1),n,1,4,0.1)

Which means, circle center (n,n) with radius 1, for all values of n between 1 and 4, going up in 0.1s – which draws a total of 31 circles!

Example 3. Suppose we make a sequence of circles with center $(10,5)$ and with radius $n$ .

Try out the following command:

Sequence[Circle[(10,5),n],n,1,5]

Example 4. Now let’s add the ‘increment parameter’ and make it 0.1, to draw many more circles.

Sequence[Circle[(10,5), n], n, 1, 5, 0.1]

The final parameter changes the increment of $n$ from the default value 1 to 0.1. More steps between 1 and 5 – more circles.

Make some of your own

Make a sequence where the center of the circle is constant, but the radius changes.
Make a sequence where the center of the circle changes, but the radius is constant.
Make a sequence where the center and the radius both change.