Lesson 4
Using Technology to Work with Sequences
Let’s use technology to create a sequence.
4.1: Where Does It Live?
Follow your teacher’s instructions to open a blank spreadsheet.
Type the following in each cell:
In A1, type 2
In A2, type 3
In A3, type -10
In A4, type 1/5
In B1, type =A1+A2
In B2, type =A3*A4
In B3, type =B1+333
In B4, type =abs(B2)
- Look at the numbers that appear in B1, B2, B3, and B4 after you press enter. Where did these numbers come from?
- Experiment with typing some different values in A1, A2, A3, and A4. Describe what happens.
- Experiment with typing some new formulas in some new cells.
- Can you figure out how to raise a number to a power?
- What happens if you forget to start a formula with the = symbol?
4.2: Fill Down
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Open a blank spreadsheet. In A1, type 10 and enter.
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In A2, type =A1+3 and enter.
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Click once on cell A2 to highlight it. See the little + in the lower-right corner? Click and drag it down to highlight several rows in that column and then let go. (This is known as “fill down.”) Describe what happens.
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What simple thing can you edit so that column A shows the sequence 12, 15, 18, . . .?
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What simple things can you edit so that column A shows the sequence 12, 11, 10, . . .?
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In B1, type 16 and enter.
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In B2, type =B1*0.5 and enter.
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Click on cell B2 and fill down. Describe what happens.
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What simple thing can you edit so that column B shows the sequence 10, 5, 2.5, . . .?
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What simple things can you edit so that column B shows the sequence 10, 30, 90, . . .?
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In column C, starting at C1 and going down, type these terms of a geometric sequence: 700, 70, 7, 0.7, 0.07
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In cell D2, type =C2/C1. What is the result?
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What is the meaning of the result?
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Click on cell D2 and fill down. What happens?
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In column E, starting at E1 and going down, type these terms of an arithmetic sequence: 7, 10.5, 14, 17.5
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In cell F2, type =E2-E1. What is the result?
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What is the meaning of the result?
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Click on cell F2 and fill down. What happens?
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Use the spreadsheet to decide whether the sequence 8, 12, 18, 27, 40.5 is arithmetic or geometric, and find its rate of change or growth factor.
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Use the spreadsheet to decide whether the sequence 50, 42.1, 34.2, 26.3 is arithmetic or geometric, and find its rate of change or growth factor.
Open a blank spreadsheet. In column A, starting in A1 and going down, enter the numbers 0 through 9. What could you type into B1 and then fill down to the 10th row that gives the first 10 terms of a linear function whose rate of change is 2 and vertical intercept is 5?
4.3: Plot Some Points
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Open a graphing utility and follow your teacher’s instructions to create a new table with 2 columns. Learn how the 2 numbers in each row can be plotted as points in the coordinate plane.
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Change the numbers in the table so that all of the plotted points lie along a diagonal line with a positive slope.
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Change the numbers in the table so that all of the plotted points lie along a horizontal line.
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Change the numbers in the table so that the graph created does not represent a function.
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Follow your teacher’s instructions to make one column a function of the other.
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Change the expression in the second column so that the plotted points lie on a line with a different steepness.
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Change the expression in the second column so that the plotted points do not lie on a line.
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Change the table so that some of the points are plotted in the second quadrant of the graph (the upper-left quadrant).
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Summary
A spreadsheet is very useful for creating new terms in a sequence. For example, for an arithmetic sequence that starts with 5 and has a rate of change of 2, you can type this. "A1" refers to "the contents of cell A1." It's like an address for the cell.
![A spreadsheet with rows 1 to 3 and column A. A 1 contains 5. A 2 contains "equals A 1 plus 2" and has a blue border. A 3 is blank.](https://staging-cms-im.s3.amazonaws.com/UxMkpKuAbq9xqJwb5sTCJYcH?response-content-disposition=inline%3B%20filename%3D%22Screenshot%202018-04-29%2010.26.14.png%22%3B%20filename%2A%3DUTF-8%27%27Screenshot%25202018-04-29%252010.26.14.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T053820Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=7393b563f6f307e58970360d6ab36cecb25068a50d835857bea670da3ae8b508)
After pressing enter, you can click cell A2 and drag the corner down to show more terms. This move is called fill down.
![A spreadsheet with rows 1 to 7 and column A. A 1 contains 5. A 2, 7. A 3, 9. A 4, 11. A 5, 13. A 6, 15. A 7, 17. A 2 through A 7 are highlighted with a blue border.](https://staging-cms-im.s3.amazonaws.com/jmg45aXRcdsk7ergeLwL2WPp?response-content-disposition=inline%3B%20filename%3D%22Screenshot%202018-04-29%2010.28.18.png%22%3B%20filename%2A%3DUTF-8%27%27Screenshot%25202018-04-29%252010.28.18.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T053820Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f9c3788afccaf02933afe6dae3ea5a7a83309386a663f76b87c9f820dd23e63e)
If you navigate to cell B2 and type =A2-A1, it will say 2. If you fill down from cell B2, all the cells will say 2. This is a way we can tell the sequence is arithmetic, and its rate of change is 2. If, instead in B2 you type =A2/A1, it will say 1.4. If you fill down from cell B2, all of the cells will contain different values. Since the sequence does not have a growth factor, we can tell the sequence is not geometric.
Technology can also be used to plot the terms of a sequence as a function of the term number. The term numbers and term values can be entered in a table and these pairs of numbers are plotted as points in the plane.
![A two column table and a graph.](https://staging-cms-im.s3.amazonaws.com/m3sn7TjDWCC8owYML7Nr96hg?response-content-disposition=inline%3B%20filename%3D%22Screenshot%202018-04-29%2010.36.23.png%22%3B%20filename%2A%3DUTF-8%27%27Screenshot%25202018-04-29%252010.36.23.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T053820Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=518818cbaa01df685beeb7c3adbd07232246b33a648f708f52513fc7d35c69b5)
You can also enter a rule for the function in the table header and automatically generate new terms. For example, here is a table and graph representing an arithmetic sequence that starts at -2 for \(x=1\) and has a rate of change of 2:
![Two column table and a discrete function graph.](https://staging-cms-im.s3.amazonaws.com/RwvV7ZP8HVCyYdZ3MRgrohuG?response-content-disposition=inline%3B%20filename%3D%22Screenshot%202018-04-29%2010.38.39.png%22%3B%20filename%2A%3DUTF-8%27%27Screenshot%25202018-04-29%252010.38.39.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T053820Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5e3be49baf2c61742380ac5ff2c90d8b2acf21e0aad515ac0aa1c90d03019c38)
Glossary Entries
- arithmetic sequence
A sequence in which each term is the previous term plus a constant.
- geometric sequence
A sequence in which each term is a constant times the previous term.