Using Tables and Graphs to Represent Geometric Sequences

4.15.2 Using Tables and Graphs to Represent Geometric Sequences

4.15.2 • Using Tables and Graphs to Represent Geometric Sequences

Activity

Clare takes a piece of paper, cuts it in half, and then stacks the pieces. She takes the stack of two pieces and cuts it in half again to form four pieces, stacking them. She keeps repeating the process.

1. The original piece of paper has length 8 inches and width 10 inches. Complete the table.

number of cuts	number of pieces	area in square inches of each piece
0
1
2
3
4
5

number of cuts	number of pieces	area in square inches of each piece
0	1	80
1	2	40
2	4	20
3	8	10
4	16	5
5	32	$\frac{5}{2}$ or 2.5

2. Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.

Compare your answer:

Double the number of pieces, from 32 to 64. Halve the area of each piece, from $\frac{5}{2}$ to $\frac{5}{4}$ .

Access multimedia content

a. Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the number of pieces as a function of the number of cuts.

A scatter plot showing the number of pieces created by 0 to 5 cuts. The number of pieces increases rapidly, reaching about 40 pieces with 5 cuts. The axes are labeled number of cuts and number of pieces.

b. How can you see on the graph how the number of pieces is changing with each cut?

Compare your answer:

The height of each plotted point is twice the height of the previous plotted point.

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a. Use the graphing tool above or technology outside the course, on the given axes, sketch a graph of the area of each piece as a function of the number of cuts.

A scatter plot shows the area of each piece (y-axis) decreasing as the number of cuts (x-axis) increases from 0 to 5. Points drop from 80 to near 0 as cuts increase, with grid lines in the background. Points are (0, 80), (1, 40), (2, 20), (3, 10), (4, 5), and (5, 2.5).

b. How can you see how the area of each piece is changing with each cut?

Compare your answer:

The height of each plotted point is half the height of the previous plotted point.

Are you ready for more?

Extending Your Thinking

1. Clare has a piece of paper that is 8 inches by 10 inches.

a. How many pieces of paper will Clare have if she cuts the paper in half $n$ times?

Compare your answer:

$2^{n}$ pieces

b. What will the area of each piece be?

Compare your answer:

Each piece has an area of $\frac{80}{2^{n}}$ square inches.

2. Why is the product of the number of pieces and the area of each piece always the same? Explain how you know.

Compare your answer:

It’s always the same because adding up the area of the little pieces of paper gives the area of the original sheet of paper.

Self Check

Given the table below, what would be the next term in the sequence?

$x$	1	2	3	4	5
$y$	270	90	30	10	$\frac{10}3$

$\frac{10}{27}$
$\frac{10}9$
$\frac{11}3$
$\frac{10}6$

Additional Resources

Graphs of Sequences

Graph the sequence {18, 36, 72, 144, 288, . . .}.

First, make a table:

Term	1	2	3	4	5
Value	18	36	72	144	288

Values for the first five terms of a geometric sequence starting with 18, where each term doubles the previous.

Now, plot the points:

A scatter plot shows points increasing rapidly, with n values 1 to 6 on the x-axis and aₙ values 0 to 360 on the y-axis, indicating a nonlinear, accelerating growth pattern.

Notice that each height is double the previous height.

Each term is also double the previous term.

Try it

Graphs of Sequences

Graph the following sequence and explain the pattern.

1, 3, 9, 27, . . .

Here is how to determine the pattern of a sequence with a graph:

First, make a table:

Term	1	2	3	4
Value	1	3	9	127

Values for the first four terms of an undefined sequence.

Next, plot the points:

Graph showing plotted points (1, 1), (2, 3), (3, 9), and (4, 27).

Each point is 3 times the height of the previous point.

Notice that in the sequence, each term is 3 times the previous term.