Sequences -- Pack Hunters -- Stretch Your Understanding

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

When a company introduces a new product they often do some preliminary market research to determine how the market might react to the product. This market research can be very extensive -- for example, they might put the product on sale at a few different stores at different prices to determine the demand at different prices or they might try different advertizing campaigns in different markets.

This set of exercises has to do with a product like a new style of clothes, whose appeal to each customer depends, in part, on its wider acceptance. We will use the sequence

q(1), q(2), ... q(n), ...

to denote weekly sales per store for weeks 1, 2, ... n, ...

The company can influence q(1) in a number of ways -- by setting the initial price, for example, or by heavy advertizing. Sometimes it is worthwhile for a company to introduce a product very "aggressively," pricing it very low and advertizing so heavily that they will lose money in the begiinning in order to gain market acceptance.

The purpose of this set of exercises is to advise a particular company regarding the introduction of a new product on the basis of their market research. The company's market research has determined that the function f(q) that determine's each week's sales on the basis of the preceding week -- that is, the function f(q) on the right hand side of the change equation

q(n) = f(q(n - 1))

is given by the graph below.



Notice two interesting features of this graph. First, if the current sales are low then next week's sales are even lower. Second, sales never seem to exceed 1,000 units per week per store.

The function shown in the graph above is defined in the CAS files for this section. You should use the appropriate CAS file to answer the questions below. The company can choose a combination of advertizing and an initial price to essentially select the first week's sales. After the first week, subsequent week's sales are determined by the model.

q(n) = f(q(n - 1))

where f(q) is the function described above and in the CAS files.

There are trade-offs that the company must consider. Although it can use very heavy advertizing and very low introductory pricing to achieve very high first week's sales, this is quite expensive.

1. What will happen if the company uses a combination of pricing and advertizing, so that the first week's sales 200 units?

2. What will happen if the company uses a combination of pricing and advertizing, so that the first week's sales 400 units?

3. What will happen if the company uses a combination of pricing and advertizing, so that the first week's sales 600 units?

4. What advice would you give to the company?