Sequences and Series

Sequences and Series

By: semba04June 1/09

Popular Ways to Solve SequenceQuestions

Excel (my favorite, so I’ll probably focus on it)

Graphing CalculatorShodor.orgBy Hand

Example: If there are 986 rooms rented out of an apartment, 12% move out and 122 move in, how many will be rented out the next

year? When the number of people level out in the apartment (when their is a constant number), how

many people will that be?We’ll do it on excel

first

First, we should (should be apparent) put the years in. There is a year 0 because we are starting with the numbers that will affect the next year.

Note: You can very easily pull down values. Click on a cell and move your cursor to the bottom right hand corner until it turns to a solid cross. Then you can pull down that box to copy the format.

The next thing I did was put a header in for the “12% move out” and I just put 122 beside the year because that’s how many move out.All the values that we are using to calculate are in a row beside the year.

As you can see I’ve inserted the equation that will calculate the amount of people that will move out (since our ‘units’ are people). Press enter and you’ll get how many people moved out.

Note: When you want to enter an equation you have to put an = in front of it.

Next I copied the format for the 122 people and the percent that moved out (used the little cross thing and dragged it to the bottom).

Then I inserted the equation that will get the number of people for out first year (we have 986 people, 12% move out and 122 move in). Note: When forming an equation and you want to use a value from another cell you can click on the cell you want the value from even if your in the middle of making an equation (you can see the dotted line around E3 – I used for my equation.)

This should be a note but, it’s important with excel. You have to be careful when copying the formatting of cells. The cell you want to copy for the total number of people is C4 because that is the cell with your equation in it. The answer for the first part of the question is 990 people (round because your not going to have half a person occupy a suite).

The second part of the question can easily be done by dragging down all your values until the number of people remain relatively constant. Note: You can’t just highlight everything you just did and drag it down , you have to go down the individual columns. The answer will be very close to 1016 people.

http://www.shodor.org/interactivate/activities/Sequencer/

Here is a clipping from Shodor.org. The values are easy enough to understand, but I find this to be limited compared to excel. If you really wanted to find out how this app. works you should go and play around with it.

You can probably tell how this would go if you did it by hand or on the home screen of you TI calculator. You would find 12% of 986. Then minus the 12% and add 122. From that answer you would repeat the process.

There is a sequence mode on your TI. I know you can start it by going to MODE and changing FUNC to SEQ. This allows you to make a graph. Pg. 117-125 in this book shows you how to make a sequence graph.

Example: In October an oil well in Alberta produced 80m³ of oil. Each month the well produces

1.3% less oil than the previous month.

A) In total, how much oil will be produced in a six month period beginning October 1?

B) Once the well production drops below 60m³ per month it is not profitable to

operate the well. In which month will it no longer be profitable to operate the

well?

The answer is on the next slide.

You can see how I bolded the sixth month. Beside that month I made an equation that adds the oil produced over the six months. The equation I used was: =sum(C19:C24)Note: All I really typed to get that equation was =sum, then I selected the cells I wanted to add. Note: You can drag down months.

To get B I dragged down all my values until the amount of oil produced dropped below 60m³. The well is not profitable after July.

Fractals

A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry.-dictionary.com

A fractal question may look like this:

Generation 1

Generation 2

Generation3 Find the perimeter

of the 4th Generation.

Here is what you might want to do:

Generation

Number of sides

Length of sides

Perimeter

1 3 27 81

2 (3*4) = 12 (27/3) = 9 108

3 (3*4)*4 = 48 (27/3)/3 = 3 144

4 ((3*4)*4)*4 = 192

((27/3)/3)/3 = 1

192

27

cm

I times the number of sides by the side length to get the perimeter of the generation.

Online fractal applications from Shodor.org:

Hilbert Curve Generator

Koch’s Snowflake

Sierpinski's Triangle

The Mandelbrot Set

Sierpinski’s Carpet

Julia Sets

And the Chaos Game:

Fun version here: