Consider This Surd As a Sequence of Terms an Where

INFINITE SURDS

The following expression is an example of an infinite surd:

Consider this surd as a sequence of terms an where:

etc.

Find a formula for an+1 in terms of an.

Calculate the decimal values of the first ten terms of the sequence. Using technology, plot the relation between n and an. Describe what you notice. What does this suggest about the value of an - an+1 as n gets very large? Use your results to find the exact value for this infinite surd.

Consider another infinite surd where the first term is . Repeat the entire process above to find the exact value for this surd.

Now consider the general infinite surd where the first term is . Find an expression for the exact value of this general infinite surd in terms of k.

The value of an infinite surd is not always an integer.

Find some values of k that make the expression an integer. Find the general statement that represents all the values of k for which the expression is an integer.

Test the validity of your general statement using other values of k.

Discuss the scope and/or limitations of your general statement.

Explain how you arrived at your general statement.

LOGARITHM BASES

Consider the following sequences. Write down the next two terms of each sequence.

Find an expression for the nth term of each sequence. Write your expressions in the form , where p, q are integers. Justify your answers using technology.

Now calculate the following, giving your answers in the form , where p, q are integers.

Describe how to obtain the third answer in each row from the first two answers. Create two more examples that fit the pattern above.

Let and . Find the general statement that expresses , in terms of cand d.

Test the validity of your general statement using other values of a, b, and x.

Discuss the scope and/or limitations of a, b, and x.

Explain how you arrived at your general statement.

PARALLELS AND PARALLELOGRAMS

This task will consider the number of parallelograms formed by intersecting parallel lines.

Figure 1 below shows a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram (A1) is formed.

A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are formed A1, A2, and A1 A2.

We can go on drawing additional transversals and forming new parallelograms.

Show that six parallelograms are formed when a fourth transversal is added to Figure 2. List all these parallelograms, using set notation.

Repeat the process with 5, 6, and 7 transversals. Show your results in a table. Use technology to find a relation between the number of transversals and the number of parallelograms. Develop a general statement, and test its validity.

Next consider the number of parallelograms formed by three horizontal parallel lines intersected by parallel transversals. Develop and test another general statement for this case.

Now extend your results to m horizontal parallel lines intersected by n parallel transversals.

Display the results in a spreadsheet and use this to find the general statement for the overall pattern.

Test the validity of your statement.

Discuss its scope and limitations.

Explain how you arrived at this generalization.